let no one destitute of geometry enter my doors” (plato ... · theorem of pythagoras in a...

Geometrical Constructios I Lecture Notes 2004 Fall Semester #01

Budapest University of Technology and Economics ◊ Faculty of Architecture ◊ Department of Architectural Representation

1

„ Let no one destitute of geometry enter mydoors” (Plato, 427-347 BC)

RAFFAELLO: School of Athens, Plato & Aristotle



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Contents

•Basic constructions

•Loci problems

•Geometrical transformations, symmetries

•Similarity, homothecy

•Regular polygons, golden ratio

•Apollonian problems

•Affine mapping, axial affinity

•Circle and ellipse

•Parabola

•Hyperbola

•Central collineation



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Drawing Instruments

For note taking and constructions

- loose-leaf white papers of the size A4

- pencils of 0.3 (H) and 0.5 (2B)

- eraser, white, soft

- set square (two triangles, 20 cm)

- pair of compasses with joint for pen

- set of color pencils

- drawing board

For drawings (assignments)

- three A2 sheets of drawing paper

- set of drawing pens (0.2, 0.4, 0.7)

- set square (two triangles, 30 cm)

- A1 sheet of drawing paper for folder

- French-curves

nameGeometrical Constructionsyear

30 30 3060

NAME

Signature GR #

I -1

Date

15

15

frame 10

folder title box

lettering



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Use of Set Squares and Pair of Compasses

perpendicularbisector

Aof a segment

copying anglebisectingan angle

parallel lines bysliding ruler

perpendicularline by rotating ruler

B



5

Triangles, some properties

A

B

C

b

c

aγ

βα

Any exterior angle of a triangle is equal to the sum of the two interior angles non adjacent tothe given exterior angle.

About the sides and opposite angles, if and only if b > a than β > α for any pair of sidesand angles opposite the sides.

Classification of triangles; acute, obtuse, right, isosceles, equilateral

Triangle inequalities

a+b > c, b+c > a, c+a > b

Sum of interior angles

α + β + γ= 180°

α+γ



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Theorems on right triangle

Theorem of Pythagoras

In a right-angled triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.

Converse:

If the sum of the squares of two sides in a triangle is equal to the square of the third side, then the triangle is a right triangle.

Theorem of Thales

In a circle, if we connect two endpoints of a diameter with any point of the circle, we get a right angle.

Converse:

If a segment AB subtends a right angle at the point C, then C is a point on the circle with diameter AB.

a2 + b2 = c2

ab

c

A B

C



7

Triangles, special points, lines and circles

O

W

C

OWC

e

P

The three altitudes of a triangle meet at a point O. This point is the orthocenter of the triangle. The orthocenter of a right triangle is the vertex at the right angle.

The three medians of a triangle meet at a point W (point of gravity). This point is the centroid of the triangle. The centroid trisect the median such that the segment connecting the vertex and the centroid is the 2/3-rd of the corresponding median.

The three perpendicular bisectors of the sides of a triangle meet at a point C. This point is the center of the circumscribed circle of the triangle.

The three points O, W, and C are collinear. The line passing through them is the Euler’s line. The ratio of the segments OW and WC is equal to 2 to 1.

The three bisectors of angles of a triangle meet at a point P. This point is the center of the inscribed circle. (Two exterior bisectors of angles and the interior bisector of the third angle also meet at a point. About this point a circle tangent to the lines of the triangle can be drawn.)



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Theorems on circles

Circles are similar

Two circles are always similar. Except the case of congruent or concentric circles, two circles have two centers of similitude C1 and C2.

Theorem on the angles at the circumference and at the center

The angle at the center is the double the angle at the circumference on the same arc.

Corollaries:

A chord subtends equal angles at the points on the same of the two arcs determined by the chord.

In a cyclic quadrilateral, the sum of the opposite angles is 180°.

Circle power

On secants of a circle passing through a point P, the product of segments is equal to the square of the tangential segment: PA1

2+PB12=PA2

2+PB22=PT2

C1

C2

C1

C2

"

""

"2"A

B

PA1

B1

A2 B2

T



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Parallel transversals, division of a segment

Theorem on parallel transversals: the ratios of the corresponding segments of arms of an angle cut by parallel lines are equal.

x:y = x’:y’, x:x’ = y:y’

xy

x’ y’

A

B

B’

C

p=3

q=5

Divide the segment AB by a point C such that AC:CB = p:q. Find the point C.

Solution: 1) draw an arbitrary ray r from A

2) measure an arbitrary unit u from A onto the rayp + q times, get the point B’

3) connect B and B’

4) draw parallel l to BB’ through the endpoint ofsegment p

5) C is the point of intersection of l and AB

u

r

l



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Quadrilaterals

Cyclic quadrilateral

"

$

" +$ =180°

Circumscribed quadrilaterala

b

c

d

a +c = b + d

M

Kite

Diagonals and bimedians

coincidence of midpoints

Trapezoid Rhombus

a a

a a



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Geometrical constructions

Euclidean construction (instruments: straight edge and a pair of compasses) :

We may fit a ruler to two given points and draw a straight line passing through them

We may measure the distance of two points by compass and draw a circle about a given point

We may determine the point of intersection of two straight lines

We may determine the points of intersection of a straight line and a circle

We may determine the points of inter section of two circles

Formal solution of constructions problems:

Sketch

Draw a sketch diagram as if the problem was solved

Plane

Try to find relations between the given data and the unknown elements, make a plane

Algorithm

Write down the algorithm of the solution

Discussion

Analyze the problem in point of view of conditions of solvability and the number of solutions



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How to solve it? (according to George Polya, 1887 - 1985)

First. You have to understand the problem.

What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory? Draw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down?

Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

Have you seen it before? Or have you seen the same problem in a slightly different form? Do you know a related problem? Do you know a theorem that could be useful? Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible? Could you restate the problem? Could you restate it still differently? Go back to definitions.



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How to solve it?

If you cannot solve the proposed problem do not let this failure afflict you too much but try to find consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again. Do not forget that human superiority consists in going around an obstacle that cannot be overcome directly, in devising some suitable auxiliary problem when the origin alone appears insoluble. Try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other? Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

Third. Carry out your plan.Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

Looking Back

Fourth. Examine the solution obtained.Can you check the result? Can you check the argument? Can you derive the solution differently? Can you see it at a glance? Can you use the result, or the method, for some other problem?



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Constructions, example

Construct parallelogram if one side an two diagonals are given. (Your solution.)

Construct parallelogram if two sides and one of the diagonals are given.

Sketch:

a

bd1

constructable ?Algorithm:

1. construct triangle with the sides of{a, b, d1}

2. reflect the triangle with respect tothe midpoint of d1

Construction:

d1 a ba

d1

b

Discussion: if the given segments satisfy the triangle-inequality, the problem is solvable.

There are two isometric solutions of oppositesense.

let no one destitute of geometry enter my doors” (plato ... · theorem of pythagoras in a...

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