Pairings of Circles and Sawayama s Theorem

Transcription

1 Forum Geometriorum Volume 13 (2013) FRUM GM SSN airings of irles and Sawayama s Teorem aris amfilos Abstrat. n tis artile we study pairs of projetively related irles, tangent to a fixed irle, and point out properties, wi in a lass of partiular ases speialize to Sawayama s teorem. 1. ntrodution Te well known Sawayama-Tebault teorem ([6]) states tat te enters 1, 2 of te two irles, defined by a evianaw of te triangleab are ollinear A 2 1 B W Figure 1. Sawayama-Tebault teorem: 1,, 2 are ollinear wit te inenter of AB (See Figure 1). Te irles, are, by definition, tangent to AW, B and te irumirle of te triangle. For te istory of te problem and a synteti proof refer to Ayme s artile [1]. n tis artile we study pairs of projetively related irles (, ) alled pairings of irles and reveal properties, tat in some partiular ases, alled Sawayama pairings, lead to Sawayama s teorem. Te general pairings of irles, onsidered ere, result by fixing a triple(,,) of a irle, a line interseting at two non-diametral points B, and a point inside te irlebut and /. Ten we onsider all lines troug te point. Te intersetion points(, ) of a line troug wit te irle, define a pair of irles (, ), wi are orrespondingly tangent to at and and also tangent to at points (, ) (See Figure 2). Tese irles ave teir enters, orrespondingly on lines ubliation ate: May 29, ommuniating ditor: aul iu.

2 118. amfilos ' ' H H B ' H' Figure 2. irle pairing defined by te triple (,,),, and it is easy to see tat teir respetive ords, pass always troug te middle of te smaller ar B, defined on by. n fat, using tis property, and onsidering a variable line troug interseting te irle at,, one an define, by orrespondingly interseting, witand defining te two isoseles triangles,. Te resulting triangles are line perspetive, sine orresponding sides interset along line. Hene tey are also point perspetive and define a point H on line, wi is ollinear wit te enters,, as well as wit te ontat points, of te irles,. Tis onstrut is alled in te sequel te basi onfiguration. n 2 we disuss a fundamental property of te basi onfiguration oming from te projetive geometry aspet. n 3 we dedue tat te line of enters always passes troug a fixed point and loate its position. n addition we sow a property of te map = f(), defined on line (Teorem 5). n 4 we investigate two penils of irles, naturally onneted to te basi onfiguration and giving te geometri meaning and onsequenes of te alulations of te preeding setion. n 5 we establis te existene of a ertain variable line, assoiated to pairing and pivoting about a fixed point (Teorem 13). Tis line, alled pivoting line, speializes in Sawayama s teorem to te evian AW from te vertex of te triangle of refereneab. n 6 we araterize te Sawayama pairings and note a resulting proof of Sawayama s teorem. Finally in 7 we supply a few details onerning te ase of Sawayama s teorem. 2. Te projetive aspet n te sequel (AB,) denotes te intersetion of te lines AB and, = (A,B) denotes te armoni onjugate of wit respet to(a,b), and(ab) = A B : A B denotes te ross ratio of te four points. Next onstrution is a typial one of te definition of a omology of te projetive plane [3, p.9]. Given two

3 airings of irles and Sawayama s teorem 119 e J H Figure 3. A typial omology = f() lines e,, interseting at point, and tree points,j, on e, define te transformation f as follows: For ea point of te plane, let be te intersetion = (,) and set = f() = (,J). efining H = (,), te equality of ross ratios(h) = (J) proves next lemma. Lemma 1. Te map f is a omology wit enter, axis and omology oeffiient equal to te ross ratiok = (J). f partiular interest for our subjet is te ase in wi pointj goes to infinity, () e () H H B Figure 4. Speializing twie te previous omology ten te onfiguration beomes like te one of figure 4() and te ross ratio as te value k =. A furter speialization of te previous figure to 4(), in wi points,, are defined by a irle (,r) and a line, interseting te irle at two non-diametral points B,, leads to te projetive aspet of our basi onfiguration. n tis ase is te projetion of te enter on and is te intersetion of te alf-line wit. Using tese notations and onventions te following lemma is valid. Lemma 2. For ea point of te irlete triangle is isoseles and te irle (, ) is tangent to te irle at and to line at. As varies on te irlete enter of desribes a parabola. n fat, te first laim follows from an easy angle asing argument (See Figure 5). From tis, or onsidering te lous as te image f() of te irle via te

4 120. amfilos A 0 N H B ' Figure 5. Tangent irles and parabola of enters omology, follows also easily tat te geometri lous of te enters of te irles is a parabola tangent to te triangle A 0 B at B, and point A 0 being te diametral of. Te parabola passes troug te middle N of A 0 and is also tangent to te irle (, B ). 3. airs of irles Te irle pairing of our basi onfiguration results by onsidering all lines troug te fixed point. Joining ea point of te irlewit and onsidering te seond intersetion point wit one defines an involution ([7, p.221]) of te points of denoted by g. oint on te irle and te orresponding H H S ' ' ' H' ' Figure 6. airing of irles for te triple(,,) point = g () define, by te reipe of te previous setion, two triangles, and teir orresponding irumirles, (See Figure6).

5 airings of irles and Sawayama s teorem 121 Lemma 3. irle is ortogonal to irles and and points,, are onyli. Tis follows by onsidering te inversion wit respet to irle. t is easily seen tat tis interanges irle and line and fixes, as a wole, irles and. Hene tese two irles are ortogonal to. Te seond laim is a onsequene of te first one. As notied in te introdution, te two triangles are perspetive and teir perspetivity enter H is on line. Wit te notations adopted so far te following is true. Lemma 4. Te lines, joining te enters of te irles,, pass troug a fixed point lying on line. Te points, are related by te omology f defined by te triple(,,): S = S. n fat, lines are images = f( ) of lines under te omology f, ene tey pass all troug = f(). As a result, lies on and taking te intersetion S = (,) we ave (S) =. Te formula results by expanding te ross ratio(s) = S : S. Teorem 5. Wit te previous notations te map = f() on line is an involution, wi in line oordinates wit origin at te projetion Q of on obtains te form, for a onstant w: z = w2 z. 0 ' S Q ' ' Figure 7. Te involution = g() n fat, using te involution = g () on te irle and interseting wit te rays of te penil troug one defines a orresponding involution = g()

6 122. amfilos on line. Tis, taking oordinatesz,z wit origin atq, is desribed by a Moebius transformation ([7,,p.157]) of te kind (See Figure7). z = az +b z a. Te rest of te teorem, on te form of tis involution (vanising of a), follows immediately from te ollinearity of,,q, wi is a onsequene of Lemma 4. n fat, if te line passing troug, obtains te position of, ten points, beome oinident and te orresponding irle beomes te tangent to at, te line of enters beomes ortogonal to and passes troug (See Figure 7). For tis partiular position of we see also tat as point onverges to, te orresponding goes to infinity. Tus for z = 0 te value of f(z) must be infinity, ene a = 0, tereby proving te teorem by setting te onstant w 2 = b/. Te value of te onstant an be easily alulated by ' 0 ' S Q ' Figure 8. Te value of te onstant letting take te plae of te intersetion point of te line wit. Ten beomes a diameter ofand, beome similar rigt angled triangles. Setting for te oordinates (e, d) and denoting te oordinates of all oter points by orresponding small letters, we obtain (z e)(z e) = d 2 and (z s)(z s) = δ, were δ = S S is te power of S wit respet to te irle. Te seond equation follows from te onyliity of points,,, (Lemma 3). From tese we obtain te value w 2 = zz = s(d2 +e 2 )+e(δ s 2 ). e s

7 airings of irles and Sawayama s teorem 123 W ' ' H Q ' Figure 9. nvolution = g() by ortogonals trougw nterpreting geometrially te teorem, we dedue tat tere is a fixed point W(0, w) on te axisq, su tat te involution = g() is defined by means of te intersetions of line by two ortogonal lines rotating about W ([5, nvolution, p.3]), tis is formulated in te following. orollary 6. Te irle pairing indues an involution = g() on line, wi oinides wit te one defined by means of pairs of ortogonal lines rotating about a fixed point W lying on lineq. 4. Two auxiliary penils of irles Here we ontinue te exploration of te basi onfiguration using te notation and onventions of te previous setions. Teorem 7. Te following properties of te basi onfiguration are true. (1) Te radial axis N of irles and passes troug and te middle N of. (2) f M = (N,Q), ten te irle m entered at M and ortogonal to and, intersets te parallel to from at two fixed points V 1 andv 2. (3) Te irles m are members of a penil wit base points V 1 and V 2. roperty (1) immediately follows from Lemma 3, sine ten is on te radial axis of te two irles, and beause of te tangeny to line, te middlen of is also on te radial axis of and (See Figure10). roperty (2) follows from te fat tat bot irles and m being ortogonal to and, teir radial axis oinides wit, wi passes troug. Te power of wit respet to and m is onstant and equal in absolute value to V 1 2, as laimed. roperty (3) is an immediate onsequene of (2).

8 124. amfilos m M H R H V 2 W V 1 ' S Q N ' ' ' H' ' Figure 10. Te penil of irles m wit base points V 1,V 2 orollary 8. Te radial axis of irle m and passes troug a fixed point R onv 1 V 2 and passes also troug H. Te first laim is a well known property for te radial axis of members of a penil and a fixed irle ([2, p.210]). Te seond laim follows from te fat tat H is te radial enter of irles, and m. Next teorem deals wit te irles aving diameter. Sine by Teorem 5 te map = g() is an involution, we know tat tis is a penil of irles ([4, vol.,p.27]). Here we make preise te kind of te penil and te loation of its base points. Note tat tis penil and te one of te preeding teorem ave a ommon member m 0 arrying te base points of bot penils. Teorem 9. Te following properties are valid. (1) Te irlen wit diameter is ortogonal to and. (2) Te irle n intersets te irle m 0 entered at Q and passing troug V 1 andv 2, at its diametral points W andw. (3) Te irles n are members of a penil wit base points W andw. roperty (1) follows from te fat tatn as its enter at te middlen of, from wi te tangents to, are equal (See Figure11). roperty (2) results from Teorem 5. Anoter proof results from te fat tat irle n belongs to te penil of ortogonal irles to and. From tis it follows tat irles m, and n pass troug te two base points U 1,U 2 lying on line, wi passes troug. Let W and W be te diametral points of irle m 0 on te diameter passing troug. Ten W W = V 1 V 2 but also, sine V 1 V 2,U 1 U 2 are ords of te irle m it is V 1 V 2 = U 1 U 2. Tis implies tat is on te radial axis ofm 0,n, wi, sine bot irles ave teir enter on, is ortogonal to. Tus it oinides wit Q or equivalently witww.

9 airings of irles and Sawayama s teorem 125 M H R m V 1 U 1 W Q N V2 U 2 ' ' ' T m 0 n ' W' Figure 11. Te penil of irles n wit base points W,W roperty (3) is an immediate onsequene of (2). Note tat point W oinides wit te one defined in orollary 6. orollary 10. Te radial axis of irle n and passes troug a fixed point T onww, and passes also troug point H. Te first laim is a well known property for te radial axis of members of a penil and a fixed irle ([2, p.210]). Te seond laim follows from te fat tat H is te radial enter of irles,m andn. 5. Te pivoting line of te pairing of irles Next figure results by drawing ortogonals respetively to lines W and W from te enters and of te irles and. Tese interset at rigt angles at point W and interset also te parallel tofrom at points W 1,W 2. Teorem 11. LinesW 1 and W 2 are parallel andw varies on a fixed line parallel to at distane equal to W. To prove te teorem a sort alulation seems unavoidable. For tis, we use oordinates along line and its ortogonal troug Q, as in 3. enoting oordinates wit respetive small letters and byr,r,r 1,r 2 respetively te radii of irles,,,, we find easily te relations zz = w 2, r 2 = 2rd, w 2 = 2d(i r) d 2 e 2, r 1 = 1 2d (2rd+(z e)2 +d 2 ), r 2 = 1 2d (2rd+(z e) 2 +d 2 ), w 1 = wz +(r 1 i)z, w 2 = wz +(r 2 i)z. w w From tese, by a sort alulation, we see tat w 2 w 1 = (z z).

10 126. amfilos V 1 W 1 S Q T W V 2 W ' ' W 2 ' '' ' ' W' n Figure 12. air of parallels W 1, W 2 Tis means tat W 1 W 2 is equal in lengt to, tus lines W 1 and W 2 are parallel, as laimed. Te oter laims are onsequenes of tis property, sine ten W 1 W 2 W and W are equal rigt angled triangles and projetingw onw tot, we get QT = W (See Figure12). Next teorem states a property of te tangents to, at pointsk,k, resulting by interseting tese irles respetively wit linesw andw. Tese tangents are te refletions of lineon linesw W 1 andw W 2 respetively. Te teorem rests upon a simple riterion reognizing te passing of a variable line troug a fixed point. ts proof is a simple alulation wi omit. x' e' q x A e Figure 13. Lines passing troug a fixed point A Lemma 12. f a variable line intersets te x axiseand a parallel to ite at distane q, so tat te oordinates x,x of te intersetion points satisfy an equation of te form ax+bx + = 0, ten te variable line passes troug te fixed point wit oordinates A = 1 a+b (,qb). Teorem 13. Te tangents to irles,, wi are refletions of, respetively, on W W 1, W W 2 are parallel and equidistant to te median W M of te rigt angled triangle W W 1 W 2. Tis median passes troug a fixed point A, independent of te position of on te irle.

11 airings of irles and Sawayama s teorem 127 V 1 W 1 S W K Q M V 2 K' ' W 2 ' ' ' T '' W ' W' n Figure 14. Te pivoting line W M and te parallel tangents at K,K Tat te tangents are parallel follows easily by measuring te angles at teir intersetion points wit. Similarly follows teir parallelity to te median W M (See Figure14).? To sow tat line W M passes troug a fixed point it suffies, aording to te previous lemma, to sow tat te x-oordinates of points M and W satisfy a linear relation. n fat, a somewat extended alulation sows tat (W ) x and M x satisfy te equation (w d)(w ) x +(2d w)m x ew = 0. From tis and te previous lemma follows tat linew M passes troug te fixed point A = 1 d (ew,i(2d w)). n te sequel linew M is alled te pivoting line of te irle pairing. 6. Sawayama pairings all te pairing of irles, defined by te basi onfiguration (,,) a Sawayama pairing, wen te orresponding point of te onfiguration is on te ar of irle lying inside(see Figure15). Te formulas of te preeding setion imply ten tat pointsw and oinide (i = w) and onsequently tatw lies on line. Tis implies in turn tat te tree parallels oinide wit te pivoting line of te pairing, wi is also te median of trianglew W 1 W 2 and wi beomes simultaneously tangent to te two irles, wile passing also troug te fixed point A. A sort alulation sows also tat in tese irumstanes point A lies on irle, tus produing te figure of Sawayama s teorem. n fat, te last ingredient of te figure is te ollinearity of points A, and, wi, in view of te previous onventions for te oordinates redues to an easy ek of te vanising of a determinant. By a well known property of te inenter ([2, p.76]),

12 128. amfilos A 2 ' ' 1 H B W S ' ' Figure 15. Sawayama pairing araterized by Q point oinides wit te inenter of triangle AB, tereby proving te next teorem. Teorem 14. a Sawayama pairing, orresponds to a triangleab wit inenter and evians AW, su tat te two irles are tangent to te irumirle of te triangle and also tangent to te evian AW and side B. Te enters, of te irles and te inenter are ollinear. Fixing te basi onfiguration(,,), so tat te orresponding point lies on te ar of ontained in, we obtain anoter proof of Sawayama s teorem, one we an sow, tat point an obtain every possible position on te aforementioned ar of and, for ea position of, te orresponding pivoting lines AW an obtain all positions of te evians troug A of triangle AB. Tat an take every position on te laimed ar of, is a simple onsequene of te omograpi relation of to desribed in 3. Next lemma makes tis point lear and sows tat te wole ar B on is obtained from a proper ar of an ellipse via te omograpy f, defined in 2. Te somewat more interesting verifiation tat te pivoting line AW obtains all positions of evians from A is andled in te next setion. Lemma 15. For varying on irle te orresponding point varies on an ellipse, wi is tangent to irle at pointsb and. n fat, sine = f() is a omology, te laimed relation is given troug te inverse omology = f 1 (), wi as te same enter and axis and ratio k =. Hene te urve = f 1 ( ) is a oni and te oter laims result from simple geometri onsiderations.

13 airings of irles and Sawayama s teorem 129 '' B S ' Figure 16. Te lous of as Q varies on 7. Te evians Using te notation of te previous setions, we dedue ere various properties of te Sawayama pairings, appearing to be of interest and leading to te proof tat, for a fixed point on te ar of lying inside, te orresponding pivoting lines AW, for variable, obtain all positions of te evians troug A. Lemma 16. Te irle wit diameter 1 2 is ortogonal to irles, and te diametral point defines line wi is parallel to B. Also te triples of points( 1,,V) and( 2,,V ) are ollinear. A radial axis V' V '' ' '' 2 R ' ' B 1 W ' ' Figure 17. Te parallel to line and te radial axis of x, Te proof of parallelity follows from te equality of anglesw 2,W 2, T 2 (See Figure 17). Te ollinearity of 1,, V follows from te equality of angles W,W and te parallelity of V 1 tow.

14 130. amfilos orollary 17. Line passes troug te intersetion points, of te irles and. '' 2 ' ' R U' k k 2 1 '' 1 H B W N U ' ' G G' Figure 18. Line W Lemma 18. Line is ortogonal to and passes troug pointw and also troug points G,G, wi are orrespondingly diametrals of wit respet to te irle wit diameter and te irle. n fat, by te previous lemma, angle is a rigt one (See Figure 18). Also is on te irle wit diameter W, wi passes troug, 1. Tus,,W are ollinear. Te enter N of te irle wit diameter is on line R, wi is te radial axis of irles, and is ortogonal to at its middle. Hene W is parallel to R. Tis implies te two oter statements of te lemma. orollary 19. Te ross ratio κ = (BW H ) is onstant and equal to UB U. HereU is te projetion of G onb. Tis follows by onsidering te penil (B,,W,H ) of lines troug. Te lines of te penil pass troug fixed points of te irle. Tus teir ross ratio is independent of te position of on tis irle. ts value is easily alulated

15 airings of irles and Sawayama s teorem 131 by letting take te plae of U, wi is te projetion of G on. Ten H goes to infinity and κ beomes equal to te stated value. orollary 20. Fixing point, wi beomes te inenter of triangle AB, te pivoting lines AW of te Sawayama pairing take te positions of all evians trougaand Sawayama s teorem is true. n fat, by te previous orollary, te onstany of ross ratio implies tat for variable onpointsh obtain all positions on lineb and onsequentlyw, being related to H by a line-omograpy, obtains also all possible positions on tis line. Referenes [1] J.-L. Ayme, Sawayama and Tebault s teorem, Forum Geom., 3 (2003) [2] N. A. ourt, ollege Geometry, over reprint, [3] L. remona, lements of rojetive Geometry, larendon ress, xford, [4] S. L. Loney, Te lements of oordinate Geometry, 2 volumes, MaMillan and o., London, [5] G. apelier, xeries de Geometrie Moderne, ditions Jaques Gabay, [6]. Sawayama, A new geometrial proposition, Amer. Mat. Montly, 12 (1905) [7]. Veblen and J. oung, rojetive Geometry, vol.,, Ginn and ompany, New ork, aris amfilos: epartment of Matematis, University of rete, rete, Greee -mail address: pamfilos@mat.uo.gr