some properties of the triangle

8/14/2019 Some Properties of the Triangle

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Some Properties of the TriangleAuthor(s): Yung-Chow-WongReviewed work(s):Source: The American Mathematical Monthly, Vol. 48, No. 8, Part I (Oct., 1941), pp. 530-535Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2303388.

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530 SOME PROPERTIES OF THE TRIANGLE [October,the correctprobability hat there s lifeon Mars. It turnsout that it is not 1/2,but 5/6. What conclusioncan you draw as to whetherthere is life on Mars?I shall leave thisquestionwithyou withoutfurther iscussion.In the second example, I have a die which s loaded so that one face is cer-tain to turnup, but I do not tell you which face this is. I am goingto make alarge numberof throwsof this die. What is the probability that I will get athreeon thefirst hrow fthe series? Since your gnorance s equally distributedamong the six faces of the die, you might conclude that the answer was 1/6.Assumingthisto be correct,what has it to do withthe resultof the first rial?And what has it to do withthe resultsof all the trials?I shall also considera variationof this problem.I have six dice in an urn.One is loaded so that the ace always turnsup; another,so that the deuce; an-other, the three; etc.A die is drawnat randomfrom he urn and thrown, ndthenreturned.This experiments repeatedmanytimes.What is the probabilityofgetting threeon a given trial?Should thisproblembe treatedin the samemanner s thepreceding ne?Again I shall leave these questionswithyouwith-out further iscussion.In conclusion, I will state that the rejectionof the statistical assumptionentails an obligationeitherto replace thisassumption by something lse whichwill serve thepurpose,or toadmitthatthetheory fprobability snot concernedwiththe resultsoftrials. f the attercourse s chosen,then thephrase theprob-abilitythat an eventwilloccur is misleading.

SOME PROPERTIES OF THE TRIANGLEYUNG-CHOW WONG,* Cambridge, Massachusetts

1. Introduction. Given arbitrarily a triangle A1A2A3, let the triangles31A2A3,tc.,t ffixed hapes be constructed n its sides.Then it is a knowntheo-rem that if AR1A2A3,tc., re all similarto an isosceles trianglewith1200vertexangle, AB1B2B3 is always equilateral,no matterwhat triangleA1A2A3 s given.Is the conversealso true? That is, if AB1B2B3 is always equilateral,no matterwhat triangleA1A2A3s given, he shapes of AB1A2A3, tc., emaining ixed, henare the triangles 1A2A3, tc.,necessarily ll similar to an isosceles trianglewith1200 vertexangle? This questiondoes not seem to have been answered. In thispaper, I solve a fewproblemsof this type,yielding ome interesting esults.Theinstrument mployed s the so-calledcomplexco6rdinatesT the methodused willbe explainedin the nextfew paragraphs.A (real) point in the plane can be represented y a single complexnumber,which s called thecomplex coordinateofthepoint.For conveniencepointswillbe denoted by capital letters nd their coordinatesby the correspondingmall

* The author is a Chinese Indemnity Funds Student.t Throughoutthispaper an expressionfollowedby etc. means thetotalityof thisexpressionand oftwo similarones obtained from t by permuting yclically the subscripts1, 2, 3.t See, for xample, Morley and Morley, Inversive Geometry.


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1941] SOME PROPERTIES OF THE TRIANGLE 531letters. Thus, whenever we speak of a point, say P, it is understood that thecoordinateof P is p; the only exception s the origin,whichhas the coordinatezero. Numbers whichare definitely nownto be real will be denoted by Greekletters.With the help of a parametriccomplexnumberri, a point B1 may be ad-joined to two arbitrarily ivenpoints A2, A3 by the following quation:(1.1) b = rl'a2 + r1a3,where(1.2) r1' +-ri = 1.In what follows t is assumed, unless statedotherwise, hat differentapital let-tersdenote distinct points,so that ri,r1' 0 or 1. Expressingri,r1' n termsofa2, 3,bi,we haveri = (b - a2)/(a3 - a2), ri' = (b - a3)/(a2- a3),

r1/r = -(b - a2)/(b - a3).Any of these equations shows that the parameterr1 determinesthe shape ofAB1A2A3, nd conversely.With this means of adjoining a point to any two points of a figure t ourdisposal, it is now clear that the problemmentionedat the beginningof thepaper can be solved by carrying ut the followingprocedures:

(i) Adjoin to the sides A2A3, etc.,of a triangleA1A2A3 the points B1, etc.,the parametriccomplexnumbers being r1, tc.(ii) Set up the conditionf(a,, a2, a3, ri, r2,r3)=0 for AB1B2B3to be equi-lateral,and equate to zero the coefficientsf the a's in it, thusobtainingthreeequations in the r's alone.(iii) If the last set ofequations are consistent, btain geometrically or ob-tain and then nterpret eometrically) ts most generalsolutionforthe r's.In thefollowing wo sections nalogous processes re carriedout,with there-spective requirements hat the triangleB1B2B3 s similar to a given triangleC1C2C3nd that the linesA1B1,etc., re concurrent.2. Some properties f a triangle.To thesides A2A3, tc., fa triangleA1A2A3let the pointsB1, etc.,be adjoined by theparametersr1, tc.;thus(2.1) bi = r1'a2+ r1a3, etc.,(2.2) r1' + r1 = 1, etc.Now the conditionthat the triangleB1B2B3be similarto a given triangleC1C2C3is

r1a2 + r1a3 c1 1(2.3) r2'a3+ r2a, c2 1 =0.r3ra1+r3a2 C3 1


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532 SOME PROPERTIES OF THE TRIANGLE [October,Equating to zero the coefficientsfa,, a2, a3 in (2.3), we obtain(2.4) r2(C3 c1) + r' (c1 -C2) = 0, etc.These can be written, ecause of (2.2),

r2(C3 - Cl) - r3(Cl - C2) = -C1 + C2, etc.The solutionsfor he r's are now readily seen to be(2.5) r2(C3 - C1) = d - cl, etc.,whered is an arbitrary omplex number. To find the geometricmeanings ofequations (2.5), we express the r's in termsof the a's and b's by means of (1.3)and their nalogous equations. We obtain(2.6) (d -C1)/(C3 - C1) = (b2- a3)/(al - a3), etc.,which are the conditionsthat the trianglesB1A2A3, tc., re (directly) imilartothetrianglesDC3C2,etc.Hence we have thefollowing:

THEOREM 2.1. If, on the ides of n arbitrarilyhosen riangle 1A2A3, rianglesB1A2A3, tc., ffixed hapes are constructed,hen n order hat hetriangleB1B2B3may always be similar to a given riangleCiC2C3,no matterwhattriangleA1A2A3is chosen, t is necessary nd sufficienthat pointD exists uch that hetrianglesB1A2A3, tc., re similar to the riangles C3C2,etc.When AC1C2C3is equilateraland ADC3C2, tc., re all similar, hen t is geo-metrically videntthatD mustbe the ncenter f AC1C2C3. his givesan answerto the questionwe asked in ?1.COROLLARY 2.1 a. If, on the sides of an arbitrarily hosen triangleA1A2A3,similartriangles 1A2A3, tc., ffixed hapes are constructed,hen n order hat hetriangle 1B2B3maybeequilateral,no matterwhattriangleA1A2A3s chosen, t isnecessarynd sufficienthat he riangles 1A2A3, tc., re isosceleswith 200 vertexanglesat theB's.It will be observed that the similar sosceles trianglesB1A2A3, tc., re con-structedall outward or all inwardaccordingas the equilateral triangleClC2C3is described n the same or opposite sense as AA1A2A3.When C1, C2, C3 are threepointson a line,we have the following:COROLLARY 2.1 b. If, onthe idesof n arbitrarilyhosen riangleA1A2A3, ri-angles B1A2A3, tc.,offixedshapes are constructed,hen n orderthat thepointsB1, B2, B3 mayalwaysbecollinear,no matterwhattriangleA1A2A3s chosen, t isnecessary nd sufficienthat ,B2A3A1= ,B3A2A1,etc.

Here `JB2A3A1denotes the directed angle* fromB2A3to A3A1, .e., the anglethroughwhichthe lineB2A3, akenas a whole,mustbe rotatedabout A3 in the* R. A. Johnson,Modern Geometry, ?16.


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1941] SOME PROPERTIES OF THE TRIANGLE 533positivedirection n order to coincidewithA3A1.From thisdefinition, irectedangles are equivalentwhenthey differ y 7r.Finally, fCl, C2, C3,D are fourpoints on a line,Theorem 2.1 gives the theo-rem ofMenelaus.

3. Some properties fa triangle continued). We nowconsidertheconditionthat the linesAlB1, etc.,be concurrent, he coordinatesofB1, etc.,beinggivenby equations (2.1). The equations ofthe linesA 1R, etc., re(- b)x - (a, - b1)? + aibi - b = 0, etc.,

where, s well as in what follows, onjugate complexnumbersare indicatedbybars; thus,forexample, d1 is the conjugate of a1. Since we are dealing with asimilitudeproperty,we may suppose without oss ofgenerality hat(3.1) al = a, a2 = O, a3=1.Then(3.2) bi = ri, b2 = r2'+ r2a, b3= r'a,and theconditionfor he concurrence fthe linesA1Bi,etc., s

- ?i + a - ri + a - dr, + a?,(3.3) ?2' + r2a r2' + r2a 0 =0.

1-r3' a 1-r3'a ?'d a- r'aEquating to zero the coefficientsf a2d, a2, ad, and a, we obtain

1(?2r3' - r2f') + r3'(r2 - ?2) = 0,?i(f r' + r2) - r3'(lr2 + ?') = 0,r1( r3' + r2) - ?'Q r2 + i' ) ?1(r2'f ' + 2) - r' (rl2 + r2') = 0,

,(r' - r2') + r3'(rlr2' - ri') = 0.The equations obtainedby equating to zero the coefficientsf d2a,a2, a are theconjugates of (3.4)1, (3.4)2, (3.4)4,* and are thereforencludedin (3.4).We shall now solve the equations (3.4), notingthat r1, 2,r3#0or 1. If oneofthe r's is real, it followsfrom 3.4), and (3.4)4 that the other two r's are alsoreal. In thiscase, equations (3.4), and (3.4)4 are identically atisfied,whileequa-tions (3.4)2 and (3.4)3 both reduce torlr2r3 = r1'r2'r'.Geometricallywe have thatB1,etc., ie respectively n A2A3, tc., ndB1A2 B2A3 B3A1 = -1.B1A3 B2A1 B3A2*Throughouthispaperweadopttheconventionhatby (3.4),wemeanthefirstquationor thefirstetof quationsn 3.4),andby 3.5) thegroup f quations onsistingf he quations(3.5)1,3.5) , *X X .


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534 SOME PROPERTIES OF THE TRIANGLE [October,Let us nowsuppose thatnone ofther's is real. Then from 3.4)1,we have

I =(3.5)3 r3 = 1,where,as we have pointed out, a Greek letteralways denotes a real number.In virtueofthis, (3.4)3 is a consequenceof (3.4)2,and (3.4), and (3.4)4 become,respectively,

rlr2- rjr2 (r2 -r2) = 0, r' -r2' + (rbr-2' i'r2) = O,i.e., r2rl'-r2ij =0, - 2' 3+?r2'r3O, which reequivalent o(3.5), rI P l- 2,

I =(3.5)2 r2 = p2r3.Writing 3.4)2 in the formrir2(1 - r3')- (1'-ri)rr3' = 0,i.e., r1r2r3-r ' r3I =0, we have, because of (3.5),(3.6) P1P2P3 = 1.Now ifwe eliminater,and r2from heequations

r3 + P3r1 = 1, fl + Plr2 = 1, r2 + p2r3 = 1,whichare equivalent to (3.5), we get

r3 + plp2p3r3 = 1-p3 + P1P3Since thep's are real but the r's are not,we musthave (3.6). Thus (3.6) is onlya consequenceof (3.5), which s thereforequivalent to (3.4).Expressingthe r's in termsofthea's and b's,we can write 3.5) in theform(3.7) (a2- b3)/(a2- a,) = p3(a2 - bl)/(d2 - a3), etc.These are the conditions that(3.8) C A1A2B3 C A3A2B1 0, etc.Hence we have thefollowing:

THEOREM 3.1. If, on thesides of an arbitrarily hosentriangleA1A2A3, ri-anglesBjA2A3, tc.,of ixed hapesare constructed,henn order hat he inesAlB1,etc.,mayalwaysbeconcurrent,omatter hat riangleA1A2A3s chosen,t s neces-saryand sufficienthat ither fthe ollowing onditions s satisfied:(i) B1, etc., ie onA2A3, tc., uch thatB1A2 B2A3 B3A1B1A3 B2A1 B3A2

(ii) CA3A1B2 gA2A B3 =O, etc.


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1941] GEOMETRY OF THE TRIANGLE IN THE KASNER PLANE 535We note that (i) gives thewell knowntheorem f Ceva, and that thesufficiencyof (ii) is also known.*In orderthatinthefigure fTheorem 2.1 the ines AdBi, tc.,may also be con-current, t is necessary nd sufficient, y Theorem3.1, that g C1C3D+ g C1C2D=0, etc.; i.e., that D is the orthocenter f triangle C1C2C3.Hence we have thefollowing:

THEOREM 3.2. If, on the sides of an arbitrarily hosen triangleA1A2A3, ri-angles 131A2A3,tc., offixedshapes are constructed,hen n order hat he riangleB1B2B3 may always be similar to a giventriangleC1C2C3and the inesA Bi, etc.,concurrent,o matterwhattriangleA1A2A3 s chosen,t is necessary nd sufficientthat g A3A1B2 g B3A1A2 -2 C2C1C3, etc.THE GEOMETRY OF THE TRIANGLE IN THE KASNER PLANE

EMANUEL MEHR, Brooklyn College1. Introduction.The aim of thispaper is to advance a symmetricnotationfor he studyofthetriangle ntheKasner plane. A new definitionfperpendicu-larity,namely quasi-perpendicularity,will be given,and new pointsof the tri-angle and theirpropertieswill be discussed.The geometry f the Kasner plane depends on the following efinitions:A point s an ordered pair of complex numbers.The distance 12from he pointPi(xi, y') to the pointP2(x2, y2), where y1-Y2,

iS (X2-X1) 2/ (y2-yl) -A line is a linear equation Ax+By+C=O with complex coefficients, hereA and B are not both zero. A line is said to be a zero ine ifA = 0, and an infinitelineifB = 0.A linear equation is said to be a general ine ifbothA and B are not zero.The slope of the general iney=px+q, (p#0), is p.Lines are parallel ifthey have no points in common. Clearly,all zero linesare parallel,all infiniteines are parallel,and twogeneral ines are parallel ftheirslopes are equal.The angle 012 rom he ineL1to the ine L2 is P2/Pl,wherepi is the slope ofLi,(i=1, 2), andp2#pl.A circle s the locus of all points at a given distance r from given point(x0,yo),togetherwith the point (x0,ye). The equation of the circle is (X-Xo)2=r(y-yo).Rigidmotion: he distance d12 nd the angle 012s defined bove are invariantunder the three-parameter roupoftransformations(1) G3: x'=mx + h, y'=m2y + k,whichmaybe called thegroupofrigidmotionsof the Kasner plane.

* Johnson,Modern Geometry, ?356.

some properties of the triangle

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