triangle 1

7/23/2019 Triangle 1

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Triangle

Triangle

A triangleEdgesand vertices 3Schlfli symbol {3} (for equilateral)

Areavarious methods;

see below

Internal angle(degrees) 60 (for equilateral)

A triangle is a polgon with three edges and three verti!es" #t is one of the basi! shapes in

geometr"A triangle with verti!esA$B$ and Cis denoted "

#n %u!lidean geometran three points$ when non&!ollinear$ determine a unique triangle and a

unique plane (i"e" a two&dimensional %u!lidean spa!e)" 'his arti!le is about triangles in

%u!lidean geometr e!ept where otherwise noted"

Contents
https://en.wikipedia.org/wiki/Edge_(geometry)https://en.wikipedia.org/wiki/Vertex_(geometry)https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbolhttps://en.wikipedia.org/wiki/Areahttps://en.wikipedia.org/wiki/Triangle#Computing_the_area_of_a_trianglehttps://en.wikipedia.org/wiki/Internal_anglehttps://en.wikipedia.org/wiki/Degree_(angle)https://en.wikipedia.org/wiki/Polygonhttps://en.wikipedia.org/wiki/Edge_(geometry)https://en.wikipedia.org/wiki/Edge_(geometry)https://en.wikipedia.org/wiki/Vertex_(geometry)https://en.wikipedia.org/wiki/Shapehttps://en.wikipedia.org/wiki/Geometryhttps://en.wikipedia.org/wiki/Geometryhttps://en.wikipedia.org/wiki/Euclidean_geometryhttps://en.wiktionary.org/wiki/collinearhttps://en.wikipedia.org/wiki/Plane_(mathematics)https://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Vertex_(geometry)https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbolhttps://en.wikipedia.org/wiki/Areahttps://en.wikipedia.org/wiki/Triangle#Computing_the_area_of_a_trianglehttps://en.wikipedia.org/wiki/Internal_anglehttps://en.wikipedia.org/wiki/Degree_(angle)https://en.wikipedia.org/wiki/Polygonhttps://en.wikipedia.org/wiki/Edge_(geometry)https://en.wikipedia.org/wiki/Vertex_(geometry)https://en.wikipedia.org/wiki/Shapehttps://en.wikipedia.org/wiki/Geometryhttps://en.wikipedia.org/wiki/Euclidean_geometryhttps://en.wiktionary.org/wiki/collinearhttps://en.wikipedia.org/wiki/Plane_(mathematics)https://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Edge_(geometry)


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'pes of triangle

o " * lengths of sides

o "+ * internal angles

+ *asi! fa!ts

o +" ,imilarit and !ongruen!e

o +"+ -ight triangles

3 %isten!e of a triangle

o 3" 'rigonometri! !onditions

. /oints$ lines$ and !ir!les asso!iated with a triangle

1omputing the sides and angles

o " 'rigonometri! ratios in right triangles

"" ,ine$ !osine and tangent

""+ #nverse fun!tions

o "+ ,ine$ !osine and tangent rules

o "3 ,olution of triangles

6 1omputing the area of a triangle

o 6" 2sing trigonometr

o 6"+ 2sing eron4s formula
https://en.wikipedia.org/wiki/Triangle#Types_of_trianglehttps://en.wikipedia.org/wiki/Triangle#By_lengths_of_sideshttps://en.wikipedia.org/wiki/Triangle#By_internal_angleshttps://en.wikipedia.org/wiki/Triangle#Basic_factshttps://en.wikipedia.org/wiki/Triangle#Basic_factshttps://en.wikipedia.org/wiki/Triangle#Similarity_and_congruencehttps://en.wikipedia.org/wiki/Triangle#Right_triangleshttps://en.wikipedia.org/wiki/Triangle#Existence_of_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Trigonometric_conditionshttps://en.wikipedia.org/wiki/Triangle#Points.2C_lines.2C_and_circles_associated_with_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Computing_the_sides_and_angleshttps://en.wikipedia.org/wiki/Triangle#Trigonometric_ratios_in_right_triangleshttps://en.wikipedia.org/wiki/Triangle#Sine.2C_cosine_and_tangenthttps://en.wikipedia.org/wiki/Triangle#Inverse_functionshttps://en.wikipedia.org/wiki/Triangle#Sine.2C_cosine_and_tangent_ruleshttps://en.wikipedia.org/wiki/Triangle#Solution_of_triangleshttps://en.wikipedia.org/wiki/Triangle#Computing_the_area_of_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Using_trigonometryhttps://en.wikipedia.org/wiki/Triangle#Using_Heron.27s_formulahttps://en.wikipedia.org/wiki/Triangle#Types_of_trianglehttps://en.wikipedia.org/wiki/Triangle#By_lengths_of_sideshttps://en.wikipedia.org/wiki/Triangle#By_internal_angleshttps://en.wikipedia.org/wiki/Triangle#Basic_factshttps://en.wikipedia.org/wiki/Triangle#Similarity_and_congruencehttps://en.wikipedia.org/wiki/Triangle#Right_triangleshttps://en.wikipedia.org/wiki/Triangle#Existence_of_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Trigonometric_conditionshttps://en.wikipedia.org/wiki/Triangle#Points.2C_lines.2C_and_circles_associated_with_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Computing_the_sides_and_angleshttps://en.wikipedia.org/wiki/Triangle#Trigonometric_ratios_in_right_triangleshttps://en.wikipedia.org/wiki/Triangle#Sine.2C_cosine_and_tangenthttps://en.wikipedia.org/wiki/Triangle#Inverse_functionshttps://en.wikipedia.org/wiki/Triangle#Sine.2C_cosine_and_tangent_ruleshttps://en.wikipedia.org/wiki/Triangle#Solution_of_triangleshttps://en.wikipedia.org/wiki/Triangle#Computing_the_area_of_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Using_trigonometryhttps://en.wikipedia.org/wiki/Triangle#Using_Heron.27s_formula


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o 6"3 2sing ve!tors

o 6". 2sing !oordinates

o 6" 2sing line integrals

o 6"6 5ormulas resembling eron4s formula

o 6" 2sing /i!74s theorem

o 6"8 9ther area formulas

o 6": 2pper bound on the area

o 6"0 *ise!ting the area

5urther formulas for general %u!lidean triangles

8 orle4s trise!tor theorem

: 5igures ins!ribed in a triangle

o :" 1oni!s

o :"+ 1onve polgon

o :"3 eagon

o :". ,quares

o :" 'riangles

0 5igures !ir!ums!ribed about a triangle

,pe!ifing the lo!ation of a point in a triangle
https://en.wikipedia.org/wiki/Triangle#Using_vectorshttps://en.wikipedia.org/wiki/Triangle#Using_coordinateshttps://en.wikipedia.org/wiki/Triangle#Using_line_integralshttps://en.wikipedia.org/wiki/Triangle#Formulas_resembling_Heron.27s_formulahttps://en.wikipedia.org/wiki/Triangle#Using_Pick.27s_theoremhttps://en.wikipedia.org/wiki/Triangle#Other_area_formulashttps://en.wikipedia.org/wiki/Triangle#Upper_bound_on_the_areahttps://en.wikipedia.org/wiki/Triangle#Bisecting_the_areahttps://en.wikipedia.org/wiki/Triangle#Further_formulas_for_general_Euclidean_triangleshttps://en.wikipedia.org/wiki/Triangle#Morley.27s_trisector_theoremhttps://en.wikipedia.org/wiki/Triangle#Figures_inscribed_in_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Figures_inscribed_in_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Conicshttps://en.wikipedia.org/wiki/Triangle#Convex_polygonhttps://en.wikipedia.org/wiki/Triangle#Hexagonhttps://en.wikipedia.org/wiki/Triangle#Squareshttps://en.wikipedia.org/wiki/Triangle#Triangleshttps://en.wikipedia.org/wiki/Triangle#Figures_circumscribed_about_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Specifying_the_location_of_a_point_in_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Using_vectorshttps://en.wikipedia.org/wiki/Triangle#Using_coordinateshttps://en.wikipedia.org/wiki/Triangle#Using_line_integralshttps://en.wikipedia.org/wiki/Triangle#Formulas_resembling_Heron.27s_formulahttps://en.wikipedia.org/wiki/Triangle#Using_Pick.27s_theoremhttps://en.wikipedia.org/wiki/Triangle#Other_area_formulashttps://en.wikipedia.org/wiki/Triangle#Upper_bound_on_the_areahttps://en.wikipedia.org/wiki/Triangle#Bisecting_the_areahttps://en.wikipedia.org/wiki/Triangle#Further_formulas_for_general_Euclidean_triangleshttps://en.wikipedia.org/wiki/Triangle#Morley.27s_trisector_theoremhttps://en.wikipedia.org/wiki/Triangle#Figures_inscribed_in_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Conicshttps://en.wikipedia.org/wiki/Triangle#Convex_polygonhttps://en.wikipedia.org/wiki/Triangle#Hexagonhttps://en.wikipedia.org/wiki/Triangle#Squareshttps://en.wikipedia.org/wiki/Triangle#Triangleshttps://en.wikipedia.org/wiki/Triangle#Figures_circumscribed_about_a_trianglehttps://en.wikipedia.org/wiki/Triangle#Specifying_the_location_of_a_point_in_a_triangle


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+ ?

Anisosceles trianglehas two sides of equal length">note ?>+?An isos!eles triangle also has

two angles of the same measure; namel$ the angles opposite to the two sides of the same

length; this fa!t is the !ontent of the isos!eles triangle theorem$ whi!h was 7nown b

%u!lid" ,ome mathemati!ians define an isos!eles triangle to have ea!tl two equal sides$

whereas others define an isos!eles triangle as one with at leasttwo equal sides">+?'he

latter definition would ma7e all equilateral triangles isos!eles triangles" 'he .@.@:0right triangle$ whi!h appears in the tetra7is square tiling$is isos!eles"

Ascalene trianglehas all its sides of different lengths">3?%quivalentl$ it has all angles of

different measure" A right triangle is also a s!alene triangle if and onl if it is not

isos!eles"
https://en.wikipedia.org/wiki/Triangle#Non-planar_triangleshttps://en.wikipedia.org/wiki/Triangle#Triangles_in_constructionhttps://en.wikipedia.org/wiki/Euler_diagramhttps://en.wikipedia.org/wiki/Equilateral_trianglehttps://en.wikipedia.org/wiki/Equilateral_trianglehttps://en.wikipedia.org/wiki/Regular_polygonhttps://en.wikipedia.org/wiki/Regular_polygonhttps://en.wikipedia.org/wiki/Triangle#cite_note-1https://en.wikipedia.org/wiki/Isosceles_trianglehttps://en.wikipedia.org/wiki/Isosceles_trianglehttps://en.wikipedia.org/wiki/Triangle#cite_note-2https://en.wikipedia.org/wiki/Triangle#cite_note-MWisosceles-3https://en.wikipedia.org/wiki/Isosceles_triangle_theoremhttps://en.wikipedia.org/wiki/Euclidhttps://en.wikipedia.org/wiki/Triangle#cite_note-MWisosceles-3https://en.wikipedia.org/wiki/Tetrakis_square_tilinghttps://en.wikipedia.org/wiki/Tetrakis_square_tilinghttps://en.wikipedia.org/wiki/Triangle#cite_note-4https://en.wikipedia.org/wiki/Triangle#cite_note-4https://en.wikipedia.org/wiki/Triangle#Non-planar_triangleshttps://en.wikipedia.org/wiki/Triangle#Triangles_in_constructionhttps://en.wikipedia.org/wiki/Euler_diagramhttps://en.wikipedia.org/wiki/Equilateral_trianglehttps://en.wikipedia.org/wiki/Regular_polygonhttps://en.wikipedia.org/wiki/Triangle#cite_note-1https://en.wikipedia.org/wiki/Isosceles_trianglehttps://en.wikipedia.org/wiki/Triangle#cite_note-2https://en.wikipedia.org/wiki/Triangle#cite_note-MWisosceles-3https://en.wikipedia.org/wiki/Isosceles_triangle_theoremhttps://en.wikipedia.org/wiki/Euclidhttps://en.wikipedia.org/wiki/Triangle#cite_note-MWisosceles-3https://en.wikipedia.org/wiki/Tetrakis_square_tilinghttps://en.wikipedia.org/wiki/Triangle#cite_note-4


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%quilateral #sos!eles ,!alene

at!h mar7s$ also !alled ti!7 mar7s$ are used in diagrams of triangles and other geometri!

figures to identif sides of equal lengths" A side !an be mar7ed with a pattern of ti!7s$ short

line segments in the form of tall mar7s; two sides have equal lengths if the are both mar7ed

with the same pattern" #n a triangle$ the pattern is usuall no more than 3 ti!7s" An equilateral

triangle has the same pattern on all 3 sides$ an isos!eles triangle has the same pattern on Bust +

sides$ and a s!alene triangle has different patterns on all sides sin!e no sides are equal" ,imilarl$

patterns of $ +$ or 3 !on!entri! ar!s inside the angles are used to indi!ate equal angles" An

equilateral triangle has the same pattern on all 3 angles$ an isos!eles triangle has the same pattern

on Bust + angles$ and a s!alene triangle has different patterns on all angles sin!e no angles are

equal"

By internal angles

'riangles !an also be !lassified a!!ording to their internal angles$ measured here in degrees"

A right triangle(or right-angled triangle$ formerl !alled a rectangled triangle) has one

of its interior angles measuring :0 (a right angle)" 'he side opposite to the right angle is

the hpotenuse$ the longest side of the triangle" 'he other two sides are !alled the legsor

catheti>.?

(singular= cathetus) of the triangle" -ight triangles obe the /thagoreantheorem= the sum of the squares of the lengths of the two legs is equal to the square of the

length of the hpotenuse= a+C b+D c+$ where aand bare the lengths of the legs and cis

the length of the hpotenuse" ,pe!ial right trianglesare right triangles with additional

properties that ma7e !al!ulations involving them easier" 9ne of the two most famous is

the 3@.@ right triangle$ where 3+C .+D +" #n this situation$ 3$ .$ and are a /thagorean
https://en.wikipedia.org/wiki/Hatch_mark#Congruency_notationhttps://en.wikipedia.org/wiki/Tally_markshttps://en.wikipedia.org/wiki/Internal_anglehttps://en.wikipedia.org/wiki/Degree_(angle)https://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Anglehttps://en.wikipedia.org/wiki/Hypotenusehttps://en.wikipedia.org/wiki/Triangle#cite_note-5https://en.wiktionary.org/wiki/cathetushttps://en.wikipedia.org/wiki/Pythagorean_theoremhttps://en.wikipedia.org/wiki/Pythagorean_theoremhttps://en.wikipedia.org/wiki/Special_right_triangleshttps://en.wikipedia.org/wiki/Pythagorean_triplehttps://en.wikipedia.org/wiki/Hatch_mark#Congruency_notationhttps://en.wikipedia.org/wiki/Tally_markshttps://en.wikipedia.org/wiki/Internal_anglehttps://en.wikipedia.org/wiki/Degree_(angle)https://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Anglehttps://en.wikipedia.org/wiki/Hypotenusehttps://en.wikipedia.org/wiki/Triangle#cite_note-5https://en.wiktionary.org/wiki/cathetushttps://en.wikipedia.org/wiki/Pythagorean_theoremhttps://en.wikipedia.org/wiki/Pythagorean_theoremhttps://en.wikipedia.org/wiki/Special_right_triangleshttps://en.wikipedia.org/wiki/Pythagorean_triple


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triple" 'he other one is an isos!eles triangle that has + angles that ea!h measure .

degrees"

'riangles that do not have an angle measuring :0 are !alled oblique triangles"

A triangle with all interior angles measuring less than :0 is an a!ute triangleor acute-

angled triangle" #f cis the length of the longest side$ then a+C b+E c+$ where aand bare

the lengths of the other sides"

A triangle with one interior angle measuring more than :0 is an obtuse triangle or

obtuse-angled triangle" #f cis the length of the longest side$ then a+C b+F c+$ where aand

bare the lengths of the other sides"

A triangle with an interior angle of 80 (and !ollinearverti!es) is degenerate"

A right degenerate triangle has !ollinear verti!es$ two of whi!h are !oin!ident"

A triangle that has two angles with the same measure also has two sides with the same length$

and therefore it is an isos!eles triangle" #t follows that in a triangle where all angles have the

same measure$ all three sides have the same length$ and su!h a triangle is therefore equilateral"

-ight 9btuse A!ute

9blique
https://en.wikipedia.org/wiki/Pythagorean_triplehttps://en.wikipedia.org/wiki/Oblique_trianglehttps://en.wikipedia.org/wiki/Acute_trianglehttps://en.wikipedia.org/wiki/Obtuse_trianglehttps://en.wiktionary.org/wiki/collinearhttps://en.wikipedia.org/wiki/Degeneracy_(mathematics)https://en.wikipedia.org/wiki/Pythagorean_triplehttps://en.wikipedia.org/wiki/Oblique_trianglehttps://en.wikipedia.org/wiki/Acute_trianglehttps://en.wikipedia.org/wiki/Obtuse_trianglehttps://en.wiktionary.org/wiki/collinearhttps://en.wikipedia.org/wiki/Degeneracy_(mathematics)


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Basic facts

A triangle$ showing eterior angle d"

'riangles are assumed to be two&dimensional plane figures$ unless the !ontet provides

otherwise (see


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measure of an eterior angle of a triangle is equal to the sum of the measures of the two interior

angles that are not adBa!ent to it; this is the eterior angle theorem" 'he sum of the measures of

the three eterior angles (one for ea!h verte) of an triangle is 360 degrees">note +?

Similarity and congruence

'wo triangles are said to besimilarif ever angle of one triangle has the same measure as the

!orresponding angle in the other triangle" 'he !orresponding sides of similar triangles have

lengths that are in the same proportion$ and this propert is also suffi!ient to establish similarit"

,ome basi! theoremsabout similar triangles are=

#f and onl ifone pair of internal angles of two triangles have the same measure as ea!hother$ and another pair also have the same measure as ea!h other$ the triangles are similar"

#f and onl if one pair of !orresponding sides of two triangles are in the same proportion

as are another pair of !orresponding sides$ and their in!luded angles have the same

measure$ then the triangles are similar" ('he included angle for an two sides of a

polgon is the internal angle between those two sides")

#f and onl if three pairs of !orresponding sides of two triangles are all in the sameproportion$ then the triangles are similar">note 3?

'wo triangles that are !ongruent have ea!tl the same siGe and shape= >note .? all pairs of

!orresponding interior angles are equal in measure$ and all pairs of !orresponding sides have the

same length" ('his is a total of si equalities$ but three are often suffi!ient to prove !ongruen!e")

,ome individuall ne!essar and suffi!ient !onditionsfor a pair of triangles to be !ongruent are=

,A, /ostulate= 'wo sides in a triangle have the same length as two sides in the other

triangle$ and the in!luded angles have the same measure"
https://en.wikipedia.org/wiki/Exterior_angle_theoremhttps://en.wikipedia.org/wiki/Triangle#cite_note-7https://en.wikipedia.org/wiki/Triangle#cite_note-7https://en.wikipedia.org/wiki/Similarity_(geometry)https://en.wikipedia.org/wiki/Theoremhttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Triangle#cite_note-8https://en.wikipedia.org/wiki/Triangle#cite_note-8https://en.wikipedia.org/wiki/Congruence_(geometry)https://en.wikipedia.org/wiki/Triangle#cite_note-9https://en.wikipedia.org/wiki/Triangle#cite_note-9https://en.wikipedia.org/wiki/Necessary_and_sufficient_conditionhttps://en.wikipedia.org/wiki/Exterior_angle_theoremhttps://en.wikipedia.org/wiki/Triangle#cite_note-7https://en.wikipedia.org/wiki/Similarity_(geometry)https://en.wikipedia.org/wiki/Theoremhttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Triangle#cite_note-8https://en.wikipedia.org/wiki/Congruence_(geometry)https://en.wikipedia.org/wiki/Triangle#cite_note-9https://en.wikipedia.org/wiki/Necessary_and_sufficient_condition


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A,A= 'wo interior angles and the in!luded side in a triangle have the same measure and

length$ respe!tivel$ as those in the other triangle" ('he included sidefor a pair of angles

is the side that is !ommon to them")

,,,= %a!h side of a triangle has the same length as a !orresponding side of the other

triangle"

AA,= 'wo angles and a !orresponding (non&in!luded) side in a triangle have the same

measure and length$ respe!tivel$ as those in the other triangle" ('his is sometimes

referred to asAAcorrSand then in!ludes A,A above")

,ome individuall suffi!ient !onditions are=

potenuse&Heg (H) 'heorem= 'he hpotenuse and a leg in a right triangle have the

same length as those in another right triangle" 'his is also !alled -, (right&angle$

hpotenuse$ side)"

potenuse&Angle 'heorem= 'he hpotenuse and an a!ute angle in one right triangle

have the same length and measure$ respe!tivel$ as those in the other right triangle" 'his

is Bust a parti!ular !ase of the AA, theorem"

An important !ondition is=

,ide&,ide&Angle (or Angle&,ide&,ide) !ondition= #f two sides and a !orresponding non&

in!luded angle of a triangle have the same length and measure$ respe!tivel$ as those in

another triangle$ then this is notsuffi!ient to prove !ongruen!e; but if the angle given is

opposite to the longer side of the two sides$ then the triangles are !ongruent" 'he

potenuse&Heg 'heorem is a parti!ular !ase of this !riterion" 'he ,ide&,ide&Angle

!ondition does not b itself guarantee that the triangles are !ongruent be!ause one

triangle !ould be obtuse&angled and the other a!ute&angled"

2sing right triangles and the !on!ept of similarit$ the trigonometri! fun!tionssine and !osine

!an be defined" 'hese are fun!tions of an anglewhi!h are investigated in trigonometr"
https://en.wikipedia.org/wiki/Trigonometric_functionhttps://en.wikipedia.org/wiki/Anglehttps://en.wikipedia.org/wiki/Trigonometryhttps://en.wikipedia.org/wiki/Trigonometric_functionhttps://en.wikipedia.org/wiki/Anglehttps://en.wikipedia.org/wiki/Trigonometry


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ight triangles

'he /thagorean theorem

A !entral theorem is the /thagorean theorem$ whi!h states in an right triangle$ the square of

the length of thehpotenuseequals the sum of the squares of the lengths of the two other sides"

#f the hpotenuse has length c$ and the legs have lengths aand b$ then the theorem states that

'he !onverse is true= if the lengths of the sides of a triangle satisf the above equation$ then the

triangle has a right angle opposite side c"

,ome other fa!ts about right triangles=

'he a!ute angles of a right triangle are !omplementar"

#f the legs of a right triangle have the same length$ then the angles opposite those legs

have the same measure" ,in!e these angles are !omplementar$ it follows that ea!h

measures . degrees" * the /thagorean theorem$ the length of the hpotenuse is thelength of a leg times I+"

#n a right triangle with a!ute angles measuring 30 and 60 degrees$ the hpotenuse is twi!e

the length of the shorter side$ and the longer side is equal to the length of the shorter side

times I3=
https://en.wikipedia.org/wiki/Pythagorean_theoremhttps://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Hypotenusehttps://en.wikipedia.org/wiki/Hypotenusehttps://en.wikipedia.org/wiki/Hypotenusehttps://en.wikipedia.org/wiki/Complementary_angleshttps://en.wikipedia.org/wiki/Pythagorean_theoremhttps://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Hypotenusehttps://en.wikipedia.org/wiki/Complementary_angles


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5or all triangles$ angles and sides are related b the law of !osinesand law of sines(also !alled

the cosine ruleandsine rule)"

E!istence of a triangle

'hetriangle inequalitstates that the sum of the lengths of an two sides of a triangle must be

greater than or equal to the length of the third side" 'hat sum !an equal the length of the third

side onl in the !ase of a degenerate triangle$ one with !ollinear verti!es" #t is not possible for

that sum to be less than the length of the third side" A triangle with three given side lengths eists

if and onl if those side lengths satisf the triangle inequalit"

'hree given angles form a non&degenerate triangle (and indeed an infinitude of them) if and onl

if both of these !onditions hold= (a) ea!h of the angles is positive$ and (b) the angles sum to 80"

#f degenerate triangles are permitted$ angles of 0 are permitted"

Trigonometric conditions

'hree positive angles $$ and $ ea!h of them less than 80$ are the angles of a triangle if and

onl ifan one of the following !onditions holds=

>6?

>6?

>?

the latter equalit appling onl if none of the angles is :0 (so the tangent fun!tion4s value is

alwas finite)"
https://en.wikipedia.org/wiki/Law_of_cosineshttps://en.wikipedia.org/wiki/Law_of_sineshttps://en.wikipedia.org/wiki/Triangle_inequalityhttps://en.wikipedia.org/wiki/Triangle_inequalityhttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Triangle#cite_note-VV-10https://en.wikipedia.org/wiki/Triangle#cite_note-VV-10https://en.wikipedia.org/wiki/Triangle#cite_note-LH-11https://en.wikipedia.org/wiki/Law_of_cosineshttps://en.wikipedia.org/wiki/Law_of_sineshttps://en.wikipedia.org/wiki/Triangle_inequalityhttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Triangle#cite_note-VV-10https://en.wikipedia.org/wiki/Triangle#cite_note-VV-10https://en.wikipedia.org/wiki/Triangle#cite_note-LH-11


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"oints# lines# and circles associated $ith a triangle

,ee also= %n!!lopedia of 'riangle 1enters

'here are thousands of different !onstru!tions that find a spe!ial point asso!iated with (and often

inside) a triangle$ satisfing some unique propert= see the referen!esse!tion for a !atalogue of

them" 9ften the are !onstru!ted b finding three lines asso!iated in a smmetri!al wa with the

three sides (or verti!es) and then proving that the three lines meet in a single point= an important

tool for proving the eisten!e of these is 1eva4s theorem$ whi!h gives a !riterion for determining

when three su!h lines are !on!urrent" ,imilarl$ lines asso!iated with a triangle are often

!onstru!ted b proving that three smmetri!all !onstru!ted points are !ollinear=here enelaus4

theorem gives a useful general !riterion" #n this se!tion Bust a few of the most !ommonlen!ountered !onstru!tions are eplained"

'he !ir!um!enteris the !enter of a !ir!le passing through the three verti!es of the triangle"

Aperpendi!ular bise!torof a side of a triangle is a straight line passing through the midpointof

the side and being perpendi!ular to it$ i"e" forming a right angle with it" 'he three perpendi!ular

bise!tors meet in a single point$ the triangle4s !ir!um!enter$ usuall denoted b %; this point isthe !enter of the !ir!um!ir!le$ the !ir!lepassing through all three verti!es" 'he diameter of this

!ir!le$ !alled the circumdiameter$ !an be found from the law of sines stated above" 'he

!ir!um!ir!le4s radius is !alled the circumradius"
https://en.wikipedia.org/wiki/Encyclopedia_of_Triangle_Centershttps://en.wikipedia.org/wiki/Triangle#Referenceshttps://en.wikipedia.org/wiki/Ceva's_theoremhttps://en.wikipedia.org/wiki/Concurrent_lineshttps://en.wiktionary.org/wiki/collinearhttps://en.wiktionary.org/wiki/collinearhttps://en.wikipedia.org/wiki/Menelaus'_theoremhttps://en.wikipedia.org/wiki/Menelaus'_theoremhttps://en.wikipedia.org/wiki/Menelaus'_theoremhttps://en.wikipedia.org/wiki/Circumcenterhttps://en.wikipedia.org/wiki/Circumcenterhttps://en.wikipedia.org/wiki/Bisectionhttps://en.wikipedia.org/wiki/Bisectionhttps://en.wikipedia.org/wiki/Midpointhttps://en.wikipedia.org/wiki/Circumcenterhttps://en.wikipedia.org/wiki/Circumcirclehttps://en.wikipedia.org/wiki/Circlehttps://en.wikipedia.org/wiki/Encyclopedia_of_Triangle_Centershttps://en.wikipedia.org/wiki/Triangle#Referenceshttps://en.wikipedia.org/wiki/Ceva's_theoremhttps://en.wikipedia.org/wiki/Concurrent_lineshttps://en.wiktionary.org/wiki/collinearhttps://en.wikipedia.org/wiki/Menelaus'_theoremhttps://en.wikipedia.org/wiki/Menelaus'_theoremhttps://en.wikipedia.org/wiki/Circumcenterhttps://en.wikipedia.org/wiki/Bisectionhttps://en.wikipedia.org/wiki/Midpointhttps://en.wikipedia.org/wiki/Circumcenterhttps://en.wikipedia.org/wiki/Circumcirclehttps://en.wikipedia.org/wiki/Circle


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'hales4 theoremimplies that if the !ir!um!enter is lo!ated on one side of the triangle$ then the

opposite angle is a right one" #f the !ir!um!enter is lo!ated inside the triangle$ then the triangle is

a!ute; if the !ir!um!enter is lo!ated outside the triangle$ then the triangle is obtuse"

'he interse!tion of the altitudes is theortho!enter"

An altitudeof a triangle is a straight line through a verte and perpendi!ular to (i"e" forming a

right angle with) the opposite side" 'his opposite side is !alled the baseof the altitude$ and the

point where the altitude interse!ts the base (or its etension) is !alled thefootof the altitude" 'he

length of the altitude is the distan!e between the base and the verte" 'he three altitudes interse!t

in a single point$ !alled the ortho!enterof the triangle$ usuall denoted b &" 'he ortho!enter lies

inside the triangle if and onl if the triangle is a!ute"

'he interse!tion of the angle bise!tors is the !enter of thein!ir!le"

An angle bise!torof a triangle is a straight line through a verte whi!h !uts the !orresponding

angle in half" 'he three angle bise!tors interse!t in a single point$ the in!enter$ usuall denoted

b I$ the !enter of the triangle4s in!ir!le" 'he in!ir!le is the !ir!le whi!h lies inside the triangle

and tou!hes all three sides" #ts radius is !alled the inradius" 'here are three other important
https://en.wikipedia.org/wiki/Thales'_theoremhttps://en.wikipedia.org/wiki/Orthocenterhttps://en.wikipedia.org/wiki/Orthocenterhttps://en.wikipedia.org/wiki/Orthocenterhttps://en.wikipedia.org/wiki/Altitude_(triangle)https://en.wikipedia.org/wiki/Orthocenterhttps://en.wikipedia.org/wiki/Incirclehttps://en.wikipedia.org/wiki/Incirclehttps://en.wikipedia.org/wiki/Angle_bisectorhttps://en.wikipedia.org/wiki/Incenterhttps://en.wikipedia.org/wiki/Incirclehttps://en.wikipedia.org/wiki/Thales'_theoremhttps://en.wikipedia.org/wiki/Orthocenterhttps://en.wikipedia.org/wiki/Altitude_(triangle)https://en.wikipedia.org/wiki/Orthocenterhttps://en.wikipedia.org/wiki/Incirclehttps://en.wikipedia.org/wiki/Angle_bisectorhttps://en.wikipedia.org/wiki/Incenterhttps://en.wikipedia.org/wiki/Incircle


14/38

!ir!les$ the e!ir!les; the lie outside the triangle and tou!h one side as well as the etensions of

the other two" 'he !enters of the in& and e!ir!les form an ortho!entri! sstem"

'he interse!tion of the medians is the !entroid"

A medianof a triangle is a straight line through a verteand the midpointof the opposite side$

and divides the triangle into two equal areas" 'he three medians interse!t in a single point$ the

triangle4s !entroid or geometri! bar!enter$ usuall denoted b '" 'he !entroid of a rigid

triangular obBe!t (!ut out of a thin sheet of uniform densit) is also its !enter of mass= the obBe!t

!an be balan!ed on its !entroid in a uniform gravitational field" 'he !entroid !uts ever median

in the ratio +=$ i"e" the distan!e between a verte and the !entroid is twi!e the distan!e between

the !entroid and the midpoint of the opposite side"


15/38

!ir!le is half that of the !ir!um!ir!le" #t tou!hes the in!ir!le (at the 5euerba!h point) and the three

e!ir!les"

%uler4s line is a straight line through the !entroid (orange)$ ortho!enter (blue)$ !ir!um!enter

(green) and !enter of the nine&point !ir!le (red)"

'he !entroid (ellow)$ ortho!enter (blue)$ !ir!um!enter (green) and !enter of the nine&point

!ir!le (red point) all lie on a single line$ 7nown as %uler4s line(red line)" 'he !enter of the nine&

point !ir!le lies at the midpoint between the ortho!enter and the !ir!um!enter$ and the distan!e

between the !entroid and the !ir!um!enter is half that between the !entroid and the ortho!enter"

'he !enter of the in!ir!le is not in general lo!ated on %uler4s line"

#f one refle!ts a median in the angle bise!tor that passes through the same verte$ one obtains a

smmedian" 'he three smmedians interse!t in a single point$ the smmedian point of the

triangle"

Computing the sides and angles

'here are various standard methods for !al!ulating the length of a side or the measure of an

angle" 1ertain methods are suited to !al!ulating values in a right&angled triangle; more !omple

methods ma be required in other situations"

Trigonometric ratios in right triangles

ain arti!le= 'rigonometri! fun!tions
https://en.wikipedia.org/wiki/Nine-point_circlehttps://en.wikipedia.org/wiki/Excirclehttps://en.wikipedia.org/wiki/Euler's_linehttps://en.wikipedia.org/wiki/Euler's_linehttps://en.wikipedia.org/wiki/Euler's_linehttps://en.wikipedia.org/wiki/Symmedianhttps://en.wikipedia.org/wiki/Symmedianhttps://en.wikipedia.org/wiki/Symmedian_pointhttps://en.wikipedia.org/wiki/Trigonometric_functionshttps://en.wikipedia.org/wiki/Nine-point_circlehttps://en.wikipedia.org/wiki/Excirclehttps://en.wikipedia.org/wiki/Euler's_linehttps://en.wikipedia.org/wiki/Euler's_linehttps://en.wikipedia.org/wiki/Symmedianhttps://en.wikipedia.org/wiki/Symmedian_pointhttps://en.wikipedia.org/wiki/Trigonometric_functions


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Aright trianglealwas in!ludes a :0 (JK+ radians) angle$ here with label 1" Angles A and * ma

var" 'rigonometri! fun!tions spe!if the relationships among side lengths and interior angles of

a right triangle"

#n right triangles$ the trigonometri! ratios of sine$ !osine and tangent !an be used to find

un7nown angles and the lengths of un7nown sides" 'he sides of the triangle are 7nown as

follows=

'he hpotenuseis the side opposite the right angle$ or defined as the longest side of a

right&angled triangle$ in this !ase h"

'he opposite sideis the side opposite to the angle we are interested in$ in this !ase a"

'he ad!acent sideis the side that is in !onta!t with the angle we are interested in and the

right angle$ hen!e its name" #n this !ase the adBa!ent side is b"

Sine# cosine and tangent

'hesineof an angle is the ratio of the length of the opposite side to the length of the hpotenuse"

#n our !ase
https://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Hypotenusehttps://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Hypotenuse


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18/38

Ar!tan !an be used to !al!ulate an angle from the length of the opposite side and the length of

the adBa!ent side"

#n introdu!tor geometr and trigonometr !ourses$ the notation sinL$ !osL$ et!"$ are often used

in pla!e of ar!sin$ ar!!os$ et!" owever$ the ar!sin$ ar!!os$ et!"$ notation is standard in higher

mathemati!s where trigonometri! fun!tions are !ommonl raised to powers$ as this avoids

!onfusion between multipli!ative inverseand !ompositional inverse"

Sine# cosine and tangent rules

ain arti!les= Haw of sines$Haw of !osinesandHaw of tangents

A triangle with sides of length a$ b and ! and angles of M$ N and O respe!tivel"

'he law of sines$or sine rule$>8?states that the ratio of the length of a side to the sine of its

!orresponding opposite angle is !onstant$ that is

'his ratio is equal to the diameter of the !ir!ums!ribed !ir!le of the given triangle" Another

interpretation of this theorem is that ever triangle with angles M$ N and O is similar to a triangle

with side lengths equal to sin M$ sin N and sin O" 'his triangle !an be !onstru!ted b first

!onstru!ting a !ir!le of diameter $ and ins!ribing in it two of the angles of the triangle" 'he

length of the sides of that triangle will be sin M$ sin N and sin O" 'he side whose length is sin M is

opposite to the angle whose measure is M$ et!"
https://en.wikipedia.org/wiki/Multiplicative_inversehttps://en.wikipedia.org/wiki/Compositional_inversehttps://en.wikipedia.org/wiki/Law_of_sineshttps://en.wikipedia.org/wiki/Law_of_cosineshttps://en.wikipedia.org/wiki/Law_of_cosineshttps://en.wikipedia.org/wiki/Law_of_cosineshttps://en.wikipedia.org/wiki/Law_of_tangentshttps://en.wikipedia.org/wiki/Law_of_tangentshttps://en.wikipedia.org/wiki/Law_of_sineshttps://en.wikipedia.org/wiki/Law_of_sineshttps://en.wikipedia.org/wiki/Triangle#cite_note-LawCosSin-12https://en.wikipedia.org/wiki/Triangle#cite_note-LawCosSin-12https://en.wikipedia.org/wiki/Multiplicative_inversehttps://en.wikipedia.org/wiki/Compositional_inversehttps://en.wikipedia.org/wiki/Law_of_sineshttps://en.wikipedia.org/wiki/Law_of_cosineshttps://en.wikipedia.org/wiki/Law_of_tangentshttps://en.wikipedia.org/wiki/Law_of_sineshttps://en.wikipedia.org/wiki/Triangle#cite_note-LawCosSin-12


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'he law of !osines$ or !osine rule$ !onne!ts the length of an un7nown side of a triangle to the

length of the other sides and the angle opposite to the un7nown side">8?As per the law=

5or a triangle with length of sides a$ b$ cand angles of M$ N$ O respe!tivel$ given two 7nown

lengths of a triangle aand b$ and the angle between the two 7nown sides O (or the angle oppositeto the un7nown side c)$ to !al!ulate the third side c$ the following formula !an be used=

#f the lengths of all three sides of an triangle are 7nown the three angles !an be !al!ulated=

'he law of tangentsor tangent rule$ !an be used to find a side or an angle when ou 7now two

sides and an angle or two angles and a side" #t states that=>:?

Solution of triangles

ain arti!le= ,olution of triangles

,olution of triangles is the histori!al term for the solving of the main trigonometri!problem= to

find missing !hara!teristi!s of a triangle (three angles$ the lengths of the three sides et!") when at

least three of these !hara!teristi!s are given" 'he triangle !an be lo!ated on a planeor on a
https://en.wikipedia.org/wiki/Law_of_cosineshttps://en.wikipedia.org/wiki/Triangle#cite_note-LawCosSin-12https://en.wikipedia.org/wiki/Law_of_tangentshttps://en.wikipedia.org/wiki/Law_of_tangentshttps://en.wikipedia.org/wiki/Triangle#cite_note-13https://en.wikipedia.org/wiki/Solution_of_triangleshttps://en.wikipedia.org/wiki/Trigonometryhttps://en.wikipedia.org/wiki/Trigonometryhttps://en.wikipedia.org/wiki/Plane_(geometry)https://en.wikipedia.org/wiki/Law_of_cosineshttps://en.wikipedia.org/wiki/Triangle#cite_note-LawCosSin-12https://en.wikipedia.org/wiki/Law_of_tangentshttps://en.wikipedia.org/wiki/Triangle#cite_note-13https://en.wikipedia.org/wiki/Solution_of_triangleshttps://en.wikipedia.org/wiki/Trigonometryhttps://en.wikipedia.org/wiki/Plane_(geometry)


20/38

sphere" 'his problem often o!!urs in various trigonometri! appli!ations$ su!h as geodes$

astronom$ !onstru!tion$ navigationet!"

Computing the area of a triangle

'he area of a triangle !an be demonstrated as half of the area of aparallelogramwhi!h has the

same base length and height"

1al!ulating the area "of a triangle is an elementar problem en!ountered often in man different

situations" 'he best 7nown and simplest formula is=

where bis the length of the base of the triangle$ and his the height or altitude of the triangle" 'he

term base denotes an side$ and height denotes the length of a perpendi!ular from the verte

opposite the side onto the line !ontaining the side itself" #n .:: 1% Arabhata$ a great

mathemati!ian&astronomerfrom the !lassi!al age of#ndian mathemati!sand#ndian astronom$

used this method in theArabhatia(se!tion +"6)"

Although simple$ this formula is onl useful if the height !an be readil found$ whi!h is not

alwas the !ase" 5or eample$ the surveor of a triangular field might find it relativel eas to

measure the length of ea!h side$ but relativel diffi!ult to !onstru!t a 4height4" Parious methods

ma be used in pra!ti!e$ depending on what is 7nown about the triangle" 'he following is a

sele!tion of frequentl used formulae for the area of a triangle">0?

sing trigonometry
https://en.wikipedia.org/wiki/Spherehttps://en.wikipedia.org/wiki/Spherehttps://en.wikipedia.org/wiki/Geodesyhttps://en.wikipedia.org/wiki/Astronomyhttps://en.wikipedia.org/wiki/Constructionhttps://en.wikipedia.org/wiki/Navigationhttps://en.wikipedia.org/wiki/Parallelogramhttps://en.wikipedia.org/wiki/Aryabhatahttps://en.wikipedia.org/wiki/Mathematicianhttps://en.wikipedia.org/wiki/Astronomerhttps://en.wikipedia.org/wiki/Indian_mathematicshttps://en.wikipedia.org/wiki/Indian_mathematicshttps://en.wikipedia.org/wiki/Indian_astronomyhttps://en.wikipedia.org/wiki/Indian_astronomyhttps://en.wikipedia.org/wiki/Aryabhatiyahttps://en.wikipedia.org/wiki/Triangle#cite_note-14https://en.wikipedia.org/wiki/Spherehttps://en.wikipedia.org/wiki/Geodesyhttps://en.wikipedia.org/wiki/Astronomyhttps://en.wikipedia.org/wiki/Constructionhttps://en.wikipedia.org/wiki/Navigationhttps://en.wikipedia.org/wiki/Parallelogramhttps://en.wikipedia.org/wiki/Aryabhatahttps://en.wikipedia.org/wiki/Mathematicianhttps://en.wikipedia.org/wiki/Astronomerhttps://en.wikipedia.org/wiki/Indian_mathematicshttps://en.wikipedia.org/wiki/Indian_astronomyhttps://en.wikipedia.org/wiki/Aryabhatiyahttps://en.wikipedia.org/wiki/Triangle#cite_note-14


21/38

Appling trigonometr to find the altitude h"

'he height of a triangle !an be found through the appli!ation of trigonometr"

#no$ing SAS= 2sing the labels in the image on the right$ the altitude is hD asin " ,ubstituting

this in the formula derived above$ the area of the triangle !an be epressed as=

(where M is the interior angle atA$ N is the interior angle atB$ is the interior angle at Cand cis

the line AB)"

5urthermore$ sin!e sin M D sin (%L M) D sin (N C )$ and similarl for the other two angles=

#no$ing AAS=

and analogousl if the 7nown side is aor c"

#no$ing ASA=>?
https://en.wikipedia.org/wiki/Trigonometryhttps://en.wikipedia.org/wiki/Triangle#cite_note-15https://en.wikipedia.org/wiki/Trigonometryhttps://en.wikipedia.org/wiki/Triangle#cite_note-15


22/38

and analogousl if the 7nown side is bor c"

sing &erons formula

'he shape of the triangle is determined b the lengths of the sides" 'herefore$ the area !an also

be derived from the lengths of the sides" *eron4s formula=

where is thesemiperimeter$ or half of the triangle4s perimeter"

'hree other equivalent was of writing eron4s formula are

sing vectors

'he area of aparallelogramembedded in a three&dimensional %u!lidean spa!e!an be !al!ulated

usingve!tors" Het ve!tors ABand ACpoint respe!tivel fromAtoBand fromAto C" 'he area

of parallelogramAB&Cis then

whi!h is the magnitude of the !ross produ!tof ve!tors ABand AC" 'he area of triangle A*1 is

half of this$
https://en.wikipedia.org/wiki/Heron's_formulahttps://en.wikipedia.org/wiki/Heron's_formulahttps://en.wikipedia.org/wiki/Parallelogramhttps://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Vector_(geometric)https://en.wikipedia.org/wiki/Vector_(geometric)https://en.wikipedia.org/wiki/Cross_producthttps://en.wikipedia.org/wiki/Cross_producthttps://en.wikipedia.org/wiki/Heron's_formulahttps://en.wikipedia.org/wiki/Parallelogramhttps://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Vector_(geometric)https://en.wikipedia.org/wiki/Cross_product


23/38

'he area of triangleABC!an also be epressed in terms ofdot produ!tsas follows=

#n two&dimensional %u!lidean spa!e$ epressing ve!tor ABas a free ve!tor in 1artesian spa!e

equal to (x$) and ACas (x+$+)$ this !an be rewritten as=

sing coordinates

#f verteAis lo!ated at the origin (0$ 0) of a 1artesian !oordinate sstemand the !oordinates of

the other two verti!es are given bBD (xB$B) and CD (xC$C)$ then the area !an be !omputed asQ+times the absolute valueof the determinant

5or three general verti!es$ the equation is=

whi!h !an be written as
https://en.wikipedia.org/wiki/Dot_producthttps://en.wikipedia.org/wiki/Dot_producthttps://en.wikipedia.org/wiki/Dot_producthttps://en.wikipedia.org/wiki/Euclidean_vector#In_Cartesian_spacehttps://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttps://en.wikipedia.org/wiki/Absolute_valuehttps://en.wikipedia.org/wiki/Determinanthttps://en.wikipedia.org/wiki/Dot_producthttps://en.wikipedia.org/wiki/Euclidean_vector#In_Cartesian_spacehttps://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttps://en.wikipedia.org/wiki/Absolute_valuehttps://en.wikipedia.org/wiki/Determinant


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#f the points are labeled sequentiall in the !ounter!lo!7wise dire!tion$ the above determinant

epressions are positive and the absolute value signs !an be omitted">+?

'he above formula is7nown as the shoela!e formulaor the surveor4s formula"

#f we lo!ate the verti!es in the !omple plane and denote them in !ounter!lo!7wise sequen!e as

aD xACAi$ bD xBCBi$ and cD xCC Ci$ and denote their !omple !onBugates as $ $ and $

then the formula

is equivalent to the shoela!e formula"

#n three dimensions$ the area of a general triangleAD (xA$A$'A)$BD (xB$B$'B) and CD (xC$C$

'C) is the /thagorean sumof the areas of the respe!tive proBe!tions on the three prin!ipal planes

(i"e"xD 0$D 0 and'D 0)=

sing line integrals

'he area within an !losed !urve$ su!h as a triangle$ is given b the line integralaround the

!urve of the algebrai! or signed distan!e of a point on the !urve from an arbitrar orientedstraight line(" /oints to the right of(as oriented are ta7en to be at negative distan!e from($

while the weight for the integral is ta7en to be the !omponent of ar! length parallel to(rather

than ar! length itself"
https://en.wikipedia.org/wiki/Triangle#cite_note-16https://en.wikipedia.org/wiki/Shoelace_formulahttps://en.wikipedia.org/wiki/Pythagorean_sumhttps://en.wikipedia.org/wiki/Line_integralhttps://en.wikipedia.org/wiki/Triangle#cite_note-16https://en.wikipedia.org/wiki/Shoelace_formulahttps://en.wikipedia.org/wiki/Pythagorean_sumhttps://en.wikipedia.org/wiki/Line_integral


25/38

'his method is well suited to !omputation of the area of an arbitrarpolgon" 'a7ing(to be the

x&ais$ the line integral between !onse!utive verti!es (xi$i) and (xiC$iC) is given b the base

times the mean height$ namel (xiCLxi)(iCiC)K+" 'he sign of the area is an overall indi!ator of

the dire!tion of traversal$ with negative area indi!ating !ounter!lo!7wise traversal" 'he area of a

triangle then falls out as the !ase of a polgon with three sides"

Rhile the line integral method has in !ommon with other !oordinate&based methods the arbitrar

!hoi!e of a !oordinate sstem$ unli7e the others it ma7es no arbitrar !hoi!e of verte of the

triangle as origin or of side as base" 5urthermore$ the !hoi!e of !oordinate sstem defined b (

!ommits to onl two degrees of freedom rather than the usual three$ sin!e the weight is a lo!al

distan!e (e"g"xiCLxiin the above) when!e the method does not require !hoosing an ais normal

to("

Rhen wor7ing inpolar !oordinatesit is not ne!essar to !onvert to !artesian !oordinatesto use

line integration$ sin!e the line integral between !onse!utive verti!es (ri$Si) and (riC$SiC) of a

polgon is given dire!tl b ririCsin(SiC L Si)K+" 'his is valid for all values of S$ with some

de!rease in numeri!al a!!ura! when TST is man orders of magnitude greater than J" Rith this

formulation negative area indi!ates !lo!7wise traversal$ whi!h should be 7ept in mind when

miing polar and !artesian !oordinates" Uust as the !hoi!e of&ais (xD 0) is immaterial for line

integration in !artesian !oordinates$ so is the !hoi!e of Gero heading (S D 0) immaterial here"

*ormulas resembling &erons formula

'hree formulas have the same stru!ture as eron4s formula but are epressed in terms of

different variables" 5irst$ denoting the medians from sides a$ b$ and crespe!tivel as ma$ mb$ and

mcand their semi&sum (maC mbC mc)K+ as V$ we have>3?

.?
https://en.wikipedia.org/wiki/Polygonhttps://en.wikipedia.org/wiki/Polar_coordinateshttps://en.wikipedia.org/wiki/Cartesian_coordinateshttps://en.wikipedia.org/wiki/Cartesian_coordinateshttps://en.wikipedia.org/wiki/Triangle#cite_note-17https://en.wikipedia.org/wiki/Triangle#cite_note-18https://en.wikipedia.org/wiki/Polygonhttps://en.wikipedia.org/wiki/Polar_coordinateshttps://en.wikipedia.org/wiki/Cartesian_coordinateshttps://en.wikipedia.org/wiki/Triangle#cite_note-17https://en.wikipedia.org/wiki/Triangle#cite_note-18


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And denoting the semi&sum of the angles4 sines as SD >(sin M) C (sin N) C (sin O)?K+$ we have>?

where&is the diameter of the !ir!um!ir!le=

sing "ic+s theorem

,ee /i!74s theoremfor a te!hnique for finding the area of an arbitrar latti!e polgon (one

drawn on a grid with verti!all and horiGontall adBa!ent latti!e points at equal distan!es$ andwith verti!es on latti!e points)"

'he theorem states=

where is the number of internal latti!e points andBis the number of latti!e points ling on the

border of the polgon"

%ther area formulas

6?
https://en.wikipedia.org/wiki/Triangle#cite_note-19https://en.wikipedia.org/wiki/Pick's_theoremhttps://en.wikipedia.org/wiki/Lattice_graphhttps://en.wikipedia.org/wiki/Inradiushttps://en.wikipedia.org/wiki/Semiperimeterhttps://en.wikipedia.org/wiki/Tangential_polygonhttps://en.wikipedia.org/wiki/Tangential_polygonhttps://en.wikipedia.org/wiki/Triangle#cite_note-20https://en.wikipedia.org/wiki/Triangle#cite_note-19https://en.wikipedia.org/wiki/Pick's_theoremhttps://en.wikipedia.org/wiki/Lattice_graphhttps://en.wikipedia.org/wiki/Inradiushttps://en.wikipedia.org/wiki/Semiperimeterhttps://en.wikipedia.org/wiki/Tangential_polygonhttps://en.wikipedia.org/wiki/Tangential_polygonhttps://en.wikipedia.org/wiki/Triangle#cite_note-20


27/38

for !ir!umdiameter&; and>?

for angle M W :0"

Xenoting the radius of the ins!ribed !ir!le as rand the radii of the e!ir!lesas r$ r+$ and r3$ the

area !an be epressed as>8?

#n 88$ *a7er>:?gave a !olle!tion of over a hundred distin!t area formulas for the triangle"

'hese in!lude=

for !ir!umradius (radius of the !ir!um!ir!le))$ and

pper bound on the area
https://en.wikipedia.org/wiki/Triangle#cite_note-21https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_trianglehttps://en.wikipedia.org/wiki/Triangle#cite_note-22https://en.wikipedia.org/wiki/Triangle#cite_note-23https://en.wikipedia.org/wiki/Triangle#cite_note-21https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_trianglehttps://en.wikipedia.org/wiki/Triangle#cite_note-22https://en.wikipedia.org/wiki/Triangle#cite_note-23


28/38

'he area of an triangle with perimeterpis less than or equal to with equalit holding if

and onl if the triangle is equilateral">+0?>+?=6

9ther upper bounds on the area "are given b>++?=p"+:0

and

both again holding if and onl if the triangle is equilateral"

Bisecting the area

'here are infinitel man lines that bise!t the area of a triangle">+3?'hree of them are the

medians$ whi!h are the onl area bise!tors that go through the !entroid" 'hree other area

bise!tors are parallel to the triangle4s sides"

An line through a triangle that splits both the triangle4s area and its perimeter in half goesthrough the triangle4s in!enter" 'here !an be one$ two$ or three of these for an given triangle"

*urther formulas for general Euclidean triangles

'he formulas in this se!tion are true for all %u!lidean triangles"

'he medians and the sides are related b>+.?=p"0

and
https://en.wikipedia.org/wiki/Triangle#cite_note-24https://en.wikipedia.org/wiki/Triangle#cite_note-25https://en.wikipedia.org/wiki/Triangle#cite_note-25https://en.wikipedia.org/wiki/Triangle#cite_note-26https://en.wikipedia.org/wiki/Triangle#cite_note-26https://en.wikipedia.org/wiki/Bisection#Area_bisectors_and_area-perimeter_bisectors_of_a_trianglehttps://en.wikipedia.org/wiki/Triangle#cite_note-Dunn-27https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-24https://en.wikipedia.org/wiki/Triangle#cite_note-25https://en.wikipedia.org/wiki/Triangle#cite_note-26https://en.wikipedia.org/wiki/Bisection#Area_bisectors_and_area-perimeter_bisectors_of_a_trianglehttps://en.wikipedia.org/wiki/Triangle#cite_note-Dunn-27https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28


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$

and equivalentl for mband mc"

5or angle M opposite side a$ the length of the internal angle bise!tor is given b

for semiperimeters$ where the bise!tor length is measured from the verte to where it meets the

opposite side"

'he interior perpendi!ular bise!tors are given b

where the sides are and the area is >+?='hm +

'he altitude from$ for eample$ the side of length ais

'he following formulas involve the !ir!umradius)and the inradius r=
https://en.wikipedia.org/wiki/Triangle#cite_note-29https://en.wikipedia.org/wiki/Triangle#cite_note-29https://en.wikipedia.org/wiki/Triangle#cite_note-29


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where haet!" are the altitudes to the subs!ripted sides;>+.?=p":

>?

and

"

,uppose two adBa!ent but non&overlapping triangles share the same side of lengthfand share the

same !ir!um!ir!le$ so that the side of lengthfis a !hord of the !ir!um!ir!le and the triangles

have side lengths (a$ b$ f) and (c$ d$ f)$ with the two triangles together forming a !!li!

quadrilateralwith side lengths in sequen!e (a$ b$ c$ d)" 'hen>+6?=8.

Het*be the !entroid of a triangle with verti!esA$B$ and C$ and let+be an interior point" 'hen

the distan!es between the points are related b>+6?=.

Hetpa$pb$ andpcbe the distan!es from the !entroid to the sides of lengths a$ b$ and c" 'hen>+6?=3

and
https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-LH-11https://en.wikipedia.org/wiki/Cyclic_quadrilateralhttps://en.wikipedia.org/wiki/Cyclic_quadrilateralhttps://en.wikipedia.org/wiki/Triangle#cite_note-Johnson-30https://en.wikipedia.org/wiki/Triangle#cite_note-Johnson-30https://en.wikipedia.org/wiki/Triangle#cite_note-Johnson-30https://en.wikipedia.org/wiki/Triangle#cite_note-Johnson-30https://en.wikipedia.org/wiki/Triangle#cite_note-Johnson-30https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-LH-11https://en.wikipedia.org/wiki/Cyclic_quadrilateralhttps://en.wikipedia.org/wiki/Cyclic_quadrilateralhttps://en.wikipedia.org/wiki/Triangle#cite_note-Johnson-30https://en.wikipedia.org/wiki/Triangle#cite_note-Johnson-30https://en.wikipedia.org/wiki/Triangle#cite_note-Johnson-30


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'he produ!t of two sides of a triangle equals the altitude to the third side times the diameter of

the !ir!um!ir!le">+.?=p"6.

1arnot4s 'heoremstates that the sum of the distan!es from the !ir!um!enter to the three sides

equals the sum of the !ir!umradius and the inradius">+.?=p"83ere a segment4s length is !onsidered

to be negative if and onl if the segment lies entirel outside the triangle" 'his method is

espe!iall useful for dedu!ing the properties of more abstra!t forms of triangles$ su!h as the ones

indu!ed b Hie algebras$ that otherwise have the same properties as usual triangles"

%uler4s theorem states that the distan!e dbetween the !ir!um!enter and the in!enter is givenb>+.?=p"8

or equivalentl

where) is the !ir!umradius and r is the inradius" 'hus for all triangles) Y +r$ with equalit

holding for equilateral triangles"

#f we denote that the ortho!enter divides one altitude into segments of lengths uand ,$ another

altitude into segment lengths $andx$ and the third altitude into segment lengthsand'$ then u,

D $xD'">+.?=p":.

'he distan!e from a side to the !ir!um!enter equals half the distan!e from the opposite verte to

the ortho!enter">+.?=p"::

'he sum of the squares of the distan!es from the verti!es to the ortho!enter plus the sum of the

squares of the sides equals twelve times the square of the !ir!umradius">+.?=p"0+
https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Carnot's_Theoremhttps://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Lie_algebrahttps://en.wikipedia.org/wiki/Euler's_theorem_in_geometryhttps://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Carnot's_Theoremhttps://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Lie_algebrahttps://en.wikipedia.org/wiki/Euler's_theorem_in_geometryhttps://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28https://en.wikipedia.org/wiki/Triangle#cite_note-Altshiller-Court-28


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5or an triangle$

,orleys trisector theorem

'he orle triangle$ resulting from the trise!tion of ea!h interior angle" 'his is an eample of afinite subdivision rule"

orle4s trise!tor theorem states that in an triangle$ the three points of interse!tion of the

adBa!ent angle trise!torsform an equilateral triangle$ !alled the orle triangle"

*igures inscribed in a triangle

Conics

As dis!ussed above$ ever triangle has a unique ins!ribed !ir!le (in!ir!le) that is interior to the

triangle and tangent to all three sides"

%ver triangle has a unique,teiner inellipsewhi!h is interior to the triangle and tangent at the

midpoints of the sides" arden4s theorem shows how to find the fo!i of this ellipse">+? 'his

ellipse has the greatest area of an ellipse tangent to all three sides of the triangle"

'he andart inellipseof a triangle is the ellipse ins!ribed within the triangle tangent to its sides

at the !onta!t points of its e!ir!les"

5or an ellipse ins!ribed in a triangleABC$ let the fo!i be+and " 'hen>+8?
https://en.wikipedia.org/wiki/Finite_subdivision_rulehttps://en.wikipedia.org/wiki/Finite_subdivision_rulehttps://en.wikipedia.org/wiki/Angle_trisectorhttps://en.wikipedia.org/wiki/Steiner_inellipsehttps://en.wikipedia.org/wiki/Steiner_inellipsehttps://en.wikipedia.org/wiki/Steiner_inellipsehttps://en.wikipedia.org/wiki/Marden's_theoremhttps://en.wikipedia.org/wiki/Ellipse#Elements_of_an_ellipsehttps://en.wikipedia.org/wiki/Triangle#cite_note-31https://en.wikipedia.org/wiki/Mandart_inellipsehttps://en.wikipedia.org/wiki/Triangle#cite_note-32https://en.wikipedia.org/wiki/Finite_subdivision_rulehttps://en.wikipedia.org/wiki/Angle_trisectorhttps://en.wikipedia.org/wiki/Steiner_inellipsehttps://en.wikipedia.org/wiki/Marden's_theoremhttps://en.wikipedia.org/wiki/Ellipse#Elements_of_an_ellipsehttps://en.wikipedia.org/wiki/Triangle#cite_note-31https://en.wikipedia.org/wiki/Mandart_inellipsehttps://en.wikipedia.org/wiki/Triangle#cite_note-32


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Conve! polygon

%ver !onve polgon with area "!an be ins!ribed in a triangle of area at most equal to +""

%qualit holds (e!lusivel) for aparallelogram">+:?

&e!agon

'he Hemoine heagonis a !!li! heagonwith verti!es given b the si interse!tions of the sides

of a triangle with the three lines that are parallel to the sides and that pass through its smmedian

point" #n either itssimple form or its self&interse!ting form$ the Hemoine heagon is interior tothe triangle with two verti!es on ea!h side of the triangle"

S-uares

%ver a!ute triangle has three ins!ribed squares (squares in its interior su!h that all four of a

square4s verti!es lie on a side of the triangle$ so two of them lie on the same side and hen!e one

side of the square !oin!ides with part of a side of the triangle)" #n a right triangle two of the

squares !oin!ide and have a verte at the triangle4s right angle$ so a right triangle has onl twodistinct ins!ribed squares" An obtuse triangle has onl one ins!ribed square$ with a side

!oin!iding with part of the triangle4s longest side" Rithin a given triangle$ a longer !ommon side

is asso!iated with a smaller ins!ribed square" #f an ins!ribed square has side of length qand the

triangle has a side of length a$ part of whi!h side !oin!ides with a side of the square$ then q$ a$

and the triangle4s area "are related a!!ording to>30?

'he largest possible ratio of the area of the ins!ribed square to the area of the triangle is K+$

whi!h o!!urs when a+D +"$ qD aK+$ and the altitude of the triangle from the base of length ais
https://en.wikipedia.org/wiki/Parallelogramhttps://en.wikipedia.org/wiki/Parallelogramhttps://en.wikipedia.org/wiki/Triangle#cite_note-33https://en.wikipedia.org/wiki/Lemoine_hexagonhttps://en.wikipedia.org/wiki/Hexagon#Cyclic_hexagonhttps://en.wikipedia.org/wiki/Symmedian_pointhttps://en.wikipedia.org/wiki/Symmedian_pointhttps://en.wikipedia.org/wiki/Polygon#Convexity_and_types_of_non-convexityhttps://en.wikipedia.org/wiki/Polygon#Convexity_and_types_of_non-convexityhttps://en.wikipedia.org/wiki/Triangle#cite_note-34https://en.wikipedia.org/wiki/Parallelogramhttps://en.wikipedia.org/wiki/Triangle#cite_note-33https://en.wikipedia.org/wiki/Lemoine_hexagonhttps://en.wikipedia.org/wiki/Hexagon#Cyclic_hexagonhttps://en.wikipedia.org/wiki/Symmedian_pointhttps://en.wikipedia.org/wiki/Symmedian_pointhttps://en.wikipedia.org/wiki/Polygon#Convexity_and_types_of_non-convexityhttps://en.wikipedia.org/wiki/Triangle#cite_note-34


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equal to a" 'he smallest possible ratio of the side of one ins!ribed square to the side of another in

the same non&obtuse triangle is >3?

Triangles

5rom an interior point in a referen!e triangle$ the nearest points on the three sides serve as the

verti!es of the pedal triangle of that point" #f the interior point is the !ir!um!enter of the

referen!e triangle$ the verti!es of the pedal triangle are the midpoints of the referen!e triangle4s

sides$ and so the pedal triangle is !alled the midpoint triangleor medial triangle" 'he midpoint

triangle subdivides the referen!e triangle into four !ongruent triangles whi!h are similar to the

referen!e triangle"

'he Zergonne triangle or intou!h triangle of a referen!e triangle has its verti!es at the three

points of tangen! of the referen!e triangle4s sides with its in!ir!le" 'he etou!h triangle of a

referen!e triangle has its verti!es at the points of tangen! of the referen!e triangle4s e!ir!les

with its sides (not etended)"

*igures circumscribed about a triangle

'he tangential triangleof a referen!e triangle (other than a right triangle) is the triangle whosesides are on the tangent linesto the referen!e triangle4s !ir!um!ir!le at its verti!es"

As mentioned above$ ever triangle has a unique !ir!um!ir!le$ a !ir!le passing through all three

verti!es$ whose !enter is the interse!tion of the perpendi!ular bise!tors of the triangle4s sides"

5urther$ ever triangle has a unique ,teiner !ir!umellipse$ whi!h passes through the triangle4s

verti!es and has its !enter at the triangle4s !entroid" 9f all ellipses going through the triangle4s

verti!es$ it has the smallest area"

'he [iepert hperbolais the unique !oni!whi!h passes through the triangle4s three verti!es$ its

!entroid$ and its !ir!um!enter"
https://en.wikipedia.org/wiki/Triangle#cite_note-35https://en.wikipedia.org/wiki/Pedal_trianglehttps://en.wikipedia.org/wiki/Midpoint_trianglehttps://en.wikipedia.org/wiki/Gergonne_trianglehttps://en.wikipedia.org/wiki/Gergonne_trianglehttps://en.wikipedia.org/wiki/Gergonne_trianglehttps://en.wikipedia.org/wiki/Extouch_trianglehttps://en.wikipedia.org/wiki/Tangential_trianglehttps://en.wikipedia.org/wiki/Tangent_linehttps://en.wikipedia.org/wiki/Tangent_linehttps://en.wikipedia.org/wiki/Steiner_ellipsehttps://en.wikipedia.org/wiki/Kiepert_hyperbolahttps://en.wikipedia.org/wiki/Conichttps://en.wikipedia.org/wiki/Triangle#cite_note-35https://en.wikipedia.org/wiki/Pedal_trianglehttps://en.wikipedia.org/wiki/Midpoint_trianglehttps://en.wikipedia.org/wiki/Gergonne_trianglehttps://en.wikipedia.org/wiki/Extouch_trianglehttps://en.wikipedia.org/wiki/Tangential_trianglehttps://en.wikipedia.org/wiki/Tangent_linehttps://en.wikipedia.org/wiki/Steiner_ellipsehttps://en.wikipedia.org/wiki/Kiepert_hyperbolahttps://en.wikipedia.org/wiki/Conic


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9f all triangles !ontained in a given !onve polgon$ there eists a triangle with maimal area

whose verti!es are all verti!es of the given polgon">3+?

Specifying the location of a point in a triangle

9ne wa to identif lo!ations of points in (or outside) a triangle is to pla!e the triangle in an

arbitrar lo!ation and orientation in the 1artesian plane$ and to use 1artesian !oordinates" Rhile

!onvenient for man purposes$ this approa!h has the disadvantage of all points4 !oordinate values

being dependent on the arbitrar pla!ement in the plane"

'wo sstems avoid that feature$ so that the !oordinates of a point are not affe!ted b moving the

triangle$ rotating it$ or refle!ting it as in a mirror$ an of whi!h give a !ongruent triangle$ or even

b res!aling it to give a similar triangle=

'rilinear !oordinates spe!if the relative distan!es of a point from the sides$ so that

!oordinates indi!ate that the ratio of the distan!e of the point from the first side

to its distan!e from the se!ond side is $ et!"

*ar!entri! !oordinatesof the form spe!if the point4s lo!ation b the relative

weights that would have to be put on the three verti!es in order to balan!e the otherwiseweightless triangle on the given point"

.on/planar triangles

A non&planar triangle is a triangle whi!h is not !ontained in a (flat) plane" ,ome eamples of

non&planar triangles in non&%u!lidean geometries are spheri!al triangles in spheri!al geometr

and hperboli! trianglesin hperboli! geometr"

Rhile the measures of the internal angles in planar triangles alwas sum to 80$ a hperboli!

triangle has measures of angles that sum to less than 80$ and a spheri!al triangle has measures

of angles that sum to more than 80" A hperboli! triangle !an be obtained b drawing on a

negativel !urved surfa!e$ su!h as a saddle surfa!e$ and a spheri!al triangle !an be obtained b

drawing on a positivel !urved surfa!e su!h as a sphere" 'hus$ if one draws a giant triangle on
https://en.wikipedia.org/wiki/Triangle#cite_note-36https://en.wikipedia.org/wiki/Cartesian_planehttps://en.wikipedia.org/wiki/Trilinear_coordinateshttps://en.wikipedia.org/wiki/Barycentric_coordinates_(mathematics)https://en.wikipedia.org/wiki/Spherical_trianglehttps://en.wikipedia.org/wiki/Spherical_geometryhttps://en.wikipedia.org/wiki/Hyperbolic_trianglehttps://en.wikipedia.org/wiki/Hyperbolic_geometryhttps://en.wikipedia.org/wiki/Saddle_surfacehttps://en.wikipedia.org/wiki/Spherehttps://en.wikipedia.org/wiki/Triangle#cite_note-36https://en.wikipedia.org/wiki/Cartesian_planehttps://en.wikipedia.org/wiki/Trilinear_coordinateshttps://en.wikipedia.org/wiki/Barycentric_coordinates_(mathematics)https://en.wikipedia.org/wiki/Spherical_trianglehttps://en.wikipedia.org/wiki/Spherical_geometryhttps://en.wikipedia.org/wiki/Hyperbolic_trianglehttps://en.wikipedia.org/wiki/Hyperbolic_geometryhttps://en.wikipedia.org/wiki/Saddle_surfacehttps://en.wikipedia.org/wiki/Sphere


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the surfa!e of the %arth$ one will find that the sum of the measures of its angles is greater than

80; in fa!t it will be between 80 and .0" >33?#n parti!ular it is possible to draw a triangle on

a sphere su!h that the measure of ea!h of its internal angles is equal to :0$ adding up to a total

of +0"

,pe!ifi!all$ on a sphere the sum of the angles of a triangle is

80 \ ( C .f)$

where f is the fra!tion of the sphere4s area whi!h is en!losed b the triangle" 5or eample$

suppose that we draw a triangle on the %arth4s surfa!e with verti!es at the


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'he 5latiron *uildingin 3.?

#n


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not easil a!!ommodate modern offi!e furniture but that has not prevented the stru!ture from

be!oming a landmar7 i!on">3?Xesigners have made houses in3:?

'riangles are sturd; while a re!tangle !an !ollapse into aparallelogramfrom pressure to one of

its points$ triangles have a natural strength whi!h supports stru!tures against lateral pressures" A

triangle will not !hange shape unless its sides are bent or etended or bro7en or if its Boints

brea7; in essen!e$ ea!h of the three sides supports the other two" A re!tangle$ in !ontrast$ is more

dependent on the strength of its Boints in a stru!tural sense" ,ome innovative designers have

proposed ma7ingbri!7snot out of re!tangles$ but with triangular shapes whi!h !an be !ombined

in three dimensions"

>.0?

#t is li7el that triangles will be used in!reasingl in new was asar!hite!ture in!reases in !ompleit" #t is important to remember that triangles are strong in

terms of rigidit$ but while pa!7ed in a tessellatingarrangement triangles are not as strong as

heagonsunder !ompression (hen!e the prevalen!e of heagonal forms in nature)" 'essellated

triangles still maintain superior strength for !antileveringhowever$ and this is the basis for one of

the strongest man made stru!tures$ the tetrahedral truss"
https://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE32-39https://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE32-39https://en.wikipedia.org/wiki/Norwayhttps://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE35-40https://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE36-41https://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE37-42https://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE41-43https://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE41-43https://en.wikipedia.org/wiki/Parallelogramhttps://en.wikipedia.org/wiki/Brickshttps://en.wikipedia.org/wiki/Brickshttps://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE33-44https://en.wikipedia.org/wiki/Tessellationhttps://en.wikipedia.org/wiki/Hexagonhttps://en.wikipedia.org/wiki/Hexagonhttps://en.wikipedia.org/wiki/Naturehttps://en.wikipedia.org/wiki/Cantileverhttps://en.wikipedia.org/wiki/Cantileverhttps://en.wikipedia.org/wiki/Space_framehttps://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE32-39https://en.wikipedia.org/wiki/Norwayhttps://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE35-40https://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE36-41https://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE37-42https://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE41-43https://en.wikipedia.org/wiki/Parallelogramhttps://en.wikipedia.org/wiki/Brickshttps://en.wikipedia.org/wiki/Triangle#cite_note-twsMarE33-44https://en.wikipedia.org/wiki/Tessellationhttps://en.wikipedia.org/wiki/Hexagonhttps://en.wikipedia.org/wiki/Naturehttps://en.wikipedia.org/wiki/Cantileverhttps://en.wikipedia.org/wiki/Space_frame

triangle 1

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