- Sawayama - Thébault's Theorem

Abstract. We present a new proof of Sawayama - Thébault's Theorem using the theory of the coaxal circles and the theory of the circular inversion.

1. Introduction

The geometer Victor Thébault (1882-1960) proposed an beautiful problem in Monthly (13) that remained unsolved several years. The first solution (14) appeared in 1973, but in 1991 a paper (15) discovered that, in 1900, a special case was treated and solved by Yûzaburô Sawayama (16). In 2003, the geometer Jean-Louis Ayme published the history of the solutions since 1973 in his article that gave a synthetic solution to the problem (17). Additionally, we encounter other solutions. In 1989, the Russian geometer Igor Federovich Sharygin (1937-2004) published a simple solution (18). Dimitrios Kodokostas published a concise and elementary solution (19). A. Ostermann and G. Wanner gave another analytic proof (20). In, 2008, Darko Veljan and Vladimir Volenec prepared other solution (21). Using Wu's method, LI Hongbo proved the theorem (22). Recently, Jean-Louis Ayme published another proof (23). The problem concerns two circles tangent to a third whose centers are collinear. This suggests a solution using the geometry of the circle - theory of the coaxal circles and the theory of the circular inversion. Thus, we elaborate a new proof using these powerful and elegant geometrical theories.

2. Main Result

Sawayama - Thébault's Theorem Let D be a point on the side BC of triangle ABC. If the circles C1 (with center P) and C2 (with center Q) are tangent to AD, BC, and also internally to the circumcircle C3 of the triangle, then the line PQ passes through the incenter I of the triangle ABC.

3. Proof of the theorem

Proof of the Sawayama - Thébault's Theorem Let R and S the points which the circumcircle C3 touch the circles C1 and C2, respectively, and let the point T the external center of similitude of the circles C1 and C2. It is easy to see that the point R is the external center of similitude of the circles C1 and C3. Similarly, the point S is the external center of similitude of the circles C2 and C3. Hence, the external centers of similitude of three circles are collinear (1, paragraph 229, page 151). R and S are antihomologous points (4). The circle C3 touch the circles C1 and C2 in like manner (2, paragraph 408, page 189). The circles C1 and C2 constitute a coaxal system of the hyperbolic type (24 or 25) whose limiting points are the points U and V. The point U lies within the circle C1 and V lies within the circle C2. The line of centers, (2, paragraph 444, page 202), of the coaxal system is the line l. Further, let E, F be the common points of circles C1 and C2, respectively, with side BC of the triangle ABC and let K and L the common points of circles C1 and C2, respectively, with the segment AD. If the points M and N are the midpoints of the segments ED and KL, respectively, the line MN is the radical axis of the coaxal system (2, paragraph 427, b, page 195). Furthermore, the points E, F and T are collinear. Indeed, the two circles have an external common tangent and this tangent pass through the external center of similitude point T of the two circles (2, paragraph 396, a, page 185). E and F are antihomologous points (1, paragraph 30, corollary, page 21). If we draw the external circle of antisimilitude C4, whose center coincides with the external center of similitude T, of the circles C1 and C2, the circles C1, C2 and C4 are coaxal (3, paragraph 114, 1, page 227). A circle of antisimilitude of two circles is a circle with regard to which they are mutually inverse. Now consider the transformation by inversion I C4 (an inversion whose circle of inversion is C4). The points E and R are the images of F and S, respectively, and the circle C3 is anallagmatic (unchanged by inversion) (5). Also the point B is the image of the C. Hence the circle C3 is orthogonal to external circle of antisimilitude. Additionally, consider the circle C5 whose center G is midpoint of arc BC of circumcircle C3 of the triangle ABC which does not contain the vertex A and that passes through vertices B and C. The inversion I (C5) carries the circumcircle C3 into side line BC (6). This transformation carries the vertex A into point X foot of the internal bisector of the angle BAC. Indeed, if only the vertex A varies on arc BC (fixed) which does not contain the point G, the inversion I C5 carries the vertex A into corresponding foot of the internal bisector of the angle BAC. The circle C6 whose diameter is the segment AX (with center Y) is orthogonal to circle C5 (10, Theorem 44, item 2, page 91). The circles C1 and C3 form a coaxal system of tangent circles whose limiting point is the point R. As the segment BC is a chord of C3 and a tangent to the C1, the angle which the chord BC subtends at limiting point is bisected by line drawn from limiting point R to the point of contact E (7). Similarly, the circles C2 and C3 form a coaxal system of tangent circles whose limiting point is the point S and therefore the segment SF is the internal bisector of the triangle BSC. Hence, by the inversion I (C5), the points E and F are the images of the points R and S, respectively, and the circles C1 and C2 are anallagmatic with regard to C5 and thus, the circle C5 belong to the conjugate coaxal system determined by coaxal system (C1 and C2) (10, Theorem 60, 1, page 130). Consequently, the circle C5 pass by limiting points U and V and the point G lies on the radical axis of coaxal system (C1 and C2). The common tangents EF and KL to the two circles C1 and C2 subtend right angles at the limiting points U and V (3, paragraph 88, corollary 3, page 178). Therefore, the points E, F, U and V are concyclic. In this circle, C7, the diameter is the segment EF. Similarly, the points K, U, L and V are concyclic. In this other circle, C8, the diameter is the segment KL. The triangles EUF and KUL are right-angled triangles whose circumcenters are the points M and N, respectively. The circumcircles of the triangles EUF and KUL are the circles C7 and C8, respectively. The circles C5, C7 and C8 form a coaxal system of the elliptic type whose basic points (2, paragraph, 445, page 202) are the points U and V. It is easy to see that the triangles EUF and KUL are equiangular and inversely similar. Indeed, if the segment FL intersect the segment KE at point W, the angles DFL, DLF and KLW are equal. Further, the angles DEK and DKE are equal. Hence, the angles EWF and KWL are equal to 90 degrees and the point W lies on the circle C7 and also at circle C8. As the point W lies within the circle C1, it coincides with the limiting point U. It is not difficult to find the axis and the center of the Dilative Reflection (9). Thus the triangle KUL is obtained from triangle EUF by a dilative reflection with center U, axis UF and similarity coefficient k=KU/UE. Consequently, the points E, U and K are collinear (perpendicular to axis of similitude UF at U) as well as the points U, L and F (axis of similitude). The segment EK is the polar of the point D with regard to circle C1 (8, paragraph 245, page 153). As the point U lies on the polar of the point D, the point D lies on the polar of the point U (8, paragraph, 247, page 154). If drawn the line f (polar of the point U with regard to circle C1), it is perpendicular to line l, line of centers of the coaxal system (C1 and C2), at point V (11, theorem 14, page 27 or 12). The circle C9 whose diameter is the segment UD pass by limiting point U and V. The line f (polar of the point U) meets the circle C5 at points V inverse of U and J, then the triangle UVJ is a right-angled triangle at V and the points U, G and J are collinear. The point G is the midpoint of UJ. The circles C1 and C6 constitute a coaxal system, but the circles C1 and C6 are anallagmatic with regard to circle C5. Hence, the circle C5 belongs to conjugate coaxal system determined by circles C1 and C6 (10, Theorem 60, page 130). The circle C5 is the common orthogonal circle of three conjugate coaxal systems determined by circles C1, C2 and C6 taken by pairs. Therefore, the point G is at radical axis of the coaxal systems formed by circles C1, C2 and C6 taken by pairs. The point G is radical center of the circles C1, C2 and C6 and, more, the circle C5 is radical circle of the three circles. If the line z is the polar of the point U with regard to circle C6, the polars of the point U with respect to the circles of the coaxal system determined by circles C1 and C6 (line f and line z) are concurrent at point Z (10, theorem 61, page 131). Similarly, the polars of the point U with respect to the circles of the coaxal system determined by circles C2 and C6 (line f and line z) are concurrent at point Z (11, theorem 22, page 71-72). The polars of the point U with respect to the circles of the coaxal system determined by circles C1 and C2 coincide with the line f because the points U and V are inverse points with regard to all circles belong to the coaxal circles determined by circles C1 and C2. The midpoint of the segment ZU is at radical axis formed by circles C1 and C6 and at radical axis formed by circles C2 and C6 (11, theorem 22, page 71-72). Additionally, segment ZU is the diameter of the common orthogonal circle belong to the conjugate coaxal systems to coaxal systems (C1 and C6) and (C2 and C6). It is easy to see that in this case the point Z coincides with the point J. The points U and J are called Polar Conjugates (11, theorem 22, page 72). If we construct the circles C10 and C11, (2, paragraph 451, page 204), circles belong to coaxal system determined by circles C1 and C6 that passes through the points U and J, respectively, the line UJ is the external common tangent to circles C10 and C11 (11, theorem 22, page 72). Similarly, if, again, construct the circles C12 and C13, circles belong to coaxal system determined by circles C6 and C2 that passes through the points U and J, respectively, the line UJ is the external common tangent to circles C12 and C13. If two circles have external common tangents, these tangents pass through the external center of similitude of the two circles (2, paragraph 396, a, page 185). Hence, the line UJ passes through external center of similitude of the circles C10 and C11. Likewise, the line UJ passes through external center of similitude of the circles C12 and C13. The external center of similitude of the circles C10 and C11 is at line of centers of coaxal system formed by circles C6 and C1. Similarly, the external center of similitude of the circles C12 and C13 is at line of centers of coaxal system formed by circles C6 and C2. The lines of centers of the two coaxal systems intersect at point Y. The external circle of antisimilitude C14, whose center is the external center of similitude, of the circles C10 and C11 is coaxal with them (1, paragraphs 126 and 127, page 96). The inversion I (C14) carries the circle C10 into circle C11. Therefore, carries the point U into point J, but if the point J is the image of the point U, the circle C13 is the image, by same transformation I (C14), of the circle C12. If the inversion I (C14) carries the circle C12 into circle C13, the circles C14, C12 and C13 are coaxal. Hence the external centers of similitude of pairs of circles, (C10 and C11) and (C12 and C13) are identical and is the common point of the lines of centers of the two coaxal systems. Clearly, the common external center of similitude is the center of the circle C6, the point Y. Therefore, the circle C6 is the common external circle of the antisimilitude of pairs circles (C10 and C11) and (C12 and C13). The two coaxal systems cannot have more than one circle in common (2, paragraph 441, page 201). The circle C14 is identical with the circle C6. The inversion I (C6) carries the point U into point J. Hence, the points Y, U, G and J are collinear. Consequently, the points U and J are at diameter AX and the points A, Y, U, X, G and J are collinear. The points U and J are the incenter I and the excenter, respectively, of the triangle ABC. Therefore, the limiting point U of the coaxal system formed by circles C1 and C2 is incenter I of the triangle ABC. The limiting points are at line of centers of the coaxal system formed by the circles C1 and C2. Hence, the points P, Q and I are collinear. If we draw the other external common tangent and the internal common tangents to circles C1 and C2, there are four triangles which satisfy the given conditions of the problem. The point U is the incenter of the two triangles and the point V is the incenter of the two others triangles. It is important to observe that the circles C1, C2 and C4 form a hyperbolic coaxal system (limiting points U and V) and its conjugate elliptic coaxal system is formed by circles C5, C7, C8, C9 and C15 (orthogonal circle of the three circles C1, C2 and C3). The circle C15 pass by points limiting points of the circles C1, C2, and C3 taken by pairs. Hence, the points R, U (or incenter of two triangles), V (or incenter of two triangles) and S are concyclic.

References

(1) Roger Arthur Johnson, Advanced Euclidean Geometry (Modern Geometry) An Elementary Treatise on the Geometry of the Triangle and the Circle, Dover, 1960.

(2) Nathan Altshiller-Court, College Geometry, An Introduction to the Modern Geometry of the Triangle and the Circle, Barnes & Noble, 1952.

(3) William J. MClelland, A Treatise on the Geometry of the Circle and some Extensions to Conic Sections by the Method of Reciprocation, Macmillan and CO, 1891.

(4) Richard Townsend, Chapters on the Modern Geometry of the Point, Line, and Circle, paragraph 209, page 288, Dublin Hodges, Smith, and CO. 1863.

(5) Julian Lowell Coolidge, A Treatise on the Circle and the Sphere, theorem 10, page 24, Oxford at the Clarendon Press, 1916.

(6) Petr Sergeevich Modenov and A. S. Parkhomenko, Geometric Transformations - volume 2 Projective Transformations, corollary I, page 115, Academic Press, 1965.

(7) John Casey, A Sequel to the First six Books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples, Book VI, Section V, Prop. 8, page 119, Dublin Hodges, Figgis, & CO. 5th ed.1888-Dublin University The Michigan Historical Reprint Series.

(8) Robert Lachlan, An Elementary Treatise on Modern Pure Geometry, London, Macmillan and CO. 1893.

(9) Isaac Moisevitch Yaglom, Geometric Transformations II, Random House, 21, theorem 2, pages 54-55, The L. W. Singer Company, New Mathematical Library, The Mathematical Association of America.

(10) Clement Vavasor Durell, Modern Geometry The Straight Line and Circle, London, Macmillan CO LTD, 1957.

(11) A. S. Smogorzhevskii, The Ruler in Geometrical Constructions, Blaisdell Publishing Company, London, 1961.

(12) Georges Papelier, Exercices de Géométrie Moderne - Précédés de L'exposé Élémentaire des Principales Théories, IV Pôles et Polaires, Théorème 1, Éditions Jacques Gabay, Paris, 1996.

(13) Victor Thébault, Problem 3887, American Mathematical Monthly, 45 (1938), pages 482-483.

(14) H. Streefkerk, Waaram eenvougig als het ook ingewikkeld kan?, Nieuw Tijdschrift voor Wiskunde 60 (1973-1973) 240-253.

(15) H. Demir and C. Tezer, Reflections on a problem of V. Thébault, Geometriae Dedicata, volume 39, number 1, pages 79-92.

(16) Y. Mikami, Mathematical Papers From the Far East, Teubner, Leipzig, 1910, page 142.

(17) Jean-Louis Ayme, Sawayama or Thébault's Theorem, Forum Geometricorum, vol. 3 (2003), pages 225-229.

(18) Igor Federovich Sharygin, Problemas de Geometria, Planimetría, Editorial Mir Moscú, Problem II. 285, pages 130, 362-363.

(19) Dimitrios Kodokostas, A Really Elementary Proof of Thébault's Theorem, Pi Mu Epsilon Journal, volume 12, page 9.

(20) Alexander Ostermann and Gerhard Wanner, A Dynamic Proof of Thébault's Theorem, Elemente der Mathematik, volume 65, issue 1, 2010, pages 12-16.

(21) Darko Veljan and Vladimir Volenec, Thébault's Theorem, Elemente der Mathematik, volume 63, issue 1, 2008, pages 6-18.

(22) LI Hongbo, Ordering in Mechanical Geometry Theorem Proving, Science in China Serie A, vol. 40, N° 3, March 1997, pages 225-233.

(23) Jean-Louis Ayme, A New Mixtilinear Incircle Adventure III, G.G.G. volume 4, http://perso.orange.frjl.ayme.

(24) OttHeinrich Keller, Analytische Geometrie Und Lineare Algebra, Veb Deutscher Verlag Der Wissenschaften, Berlin, 1957, page 280.

(25) Isaac Moisevitch Yaglom, Geometric Transformations IV, Anneli Lax New Mathematical Library, #44, Mathematical Association of America, pages 56 and 60.

Sincerely,

Deoclecio Gouveia Mota Junior. - --- In Hyacinthos@yahoogroups.com, "Deoclecio" <deocleciomota@...> wrote:
>

Dear Deoclecio,

> Sawayama - Thébault's Theorem

>

> Abstract. We present a new proof of Sawayama - Thébault's Theorem using the theory of the coaxal circles and the theory of the circular inversion.

> ..............

In H. msg# 15996 I mention the fact that incenter is a limiting point

of Thebault circles, and in # 18944 ask for ,if possible, a simple

proof of this fact.

There were no responses.

Maybe the status V.Thebault achieved in geometry

over these years makes people reluctant to look

for an easy proof of the above limiting point fact,

for it will make collinearity part of his theorem trivial

and the charm of story associated with his problem dim somewhat.

Well, it was hard to read your paper, even though I had an additional

interest in your attempt to prove the limiting point - incenter coincidence.

I gave up after reading about half.

Maybe if you clarify and separate logical steps

it will become more acceptable.

Still, yours seems to me to be a 'heavy' proof at first and

partial glance.

Mark T. - --- In Hyacinthos@yahoogroups.com, "armpist" <armpist@...> wrote:
>

Dear Mark,

>

>

> --- In Hyacinthos@yahoogroups.com, "Deoclecio" <deocleciomota@> wrote:

> >

> > Sawayama - Thébault's Theorem

> >

> > Abstract. We present a new proof of Sawayama - Thébault's Theorem using the theory of the coaxal circles and the theory of the circular inversion.

> > ..............

>

>

> Dear Deoclecio,

>

> In H. msg# 15996 I mention the fact that incenter is a limiting point

> of Thebault circles, and in # 18944 ask for ,if possible, a simple

> proof of this fact.

>

> There were no responses.

>

> Maybe the status V.Thebault achieved in geometry

> over these years makes people reluctant to look

> for an easy proof of the above limiting point fact,

> for it will make collinearity part of his theorem trivial

> and the charm of story associated with his problem dim somewhat.

>

>

> Well, it was hard to read your paper, even though I had an additional

> interest in your attempt to prove the limiting point - incenter coincidence.

>

> I gave up after reading about half.

>

> Maybe if you clarify and separate logical steps

> it will become more acceptable.

>

> Still, yours seems to me to be a 'heavy' proof at first and

> partial glance.

>

>

> Mark T.

My paper demonstrates that the limiting point of the Thebault Circles is the incenter of the basic triangle. The proof is based on the geometry of the circle. This geometry is heavy but the references can guide you to the correct understanding.

This proof was obtained in 2005 and now published.

Basically, I work with the circles coaxal (the two thebault circles) and apply the theory of inversion. If you read calmly and follow the references, you will conclude that the proof is correct.

I can not see an easier synthetic proof for the fact that the incenter is the limiting point.

Sincerely,

Deoclecio Gouveia Mota Junior - Dear Deoclecio Gouveia Mota Junior.

I read the different message and ask you if you can put on this site or send me the complete article that you have presented to Forum Geometricorum. I think that what I read on Hyacinthos is just a reduction of your paper.

Thank in advance

Sincerely

Jean-Louis

--- In Hyacinthos@yahoogroups.com, "Deoclecio" <deocleciomota@...> wrote:

>

> Sawayama - Thébault's Theorem

>

> Abstract. We present a new proof of Sawayama - Thébault's Theorem using the theory of the coaxal circles and the theory of the circular inversion.

>

> 1. Introduction

>

> The geometer Victor Thébault (1882-1960) proposed an beautiful problem in Monthly (13) that remained unsolved several years. The first solution (14) appeared in 1973, but in 1991 a paper (15) discovered that, in 1900, a special case was treated and solved by Yûzaburô Sawayama (16). In 2003, the geometer Jean-Louis Ayme published the history of the solutions since 1973 in his article that gave a synthetic solution to the problem (17). Additionally, we encounter other solutions. In 1989, the Russian geometer Igor Federovich Sharygin (1937-2004) published a simple solution (18). Dimitrios Kodokostas published a concise and elementary solution (19). A. Ostermann and G. Wanner gave another analytic proof (20). In, 2008, Darko Veljan and Vladimir Volenec prepared other solution (21). Using Wu's method, LI Hongbo proved the theorem (22). Recently, Jean-Louis Ayme published another proof (23). The problem concerns two circles tangent to a third whose centers are collinear. This suggests a solution using the geometry of the circle - theory of the coaxal circles and the theory of the circular inversion. Thus, we elaborate a new proof using these powerful and elegant geometrical theories.

>

> 2. Main Result

>

> Sawayama - Thébault's Theorem Let D be a point on the side BC of triangle ABC. If the circles C1 (with center P) and C2 (with center Q) are tangent to AD, BC, and also internally to the circumcircle C3 of the triangle, then the line PQ passes through the incenter I of the triangle ABC.

>

> 3. Proof of the theorem

>

> Proof of the Sawayama - Thébault's Theorem Let R and S the points which the circumcircle C3 touch the circles C1 and C2, respectively, and let the point T the external center of similitude of the circles C1 and C2. It is easy to see that the point R is the external center of similitude of the circles C1 and C3. Similarly, the point S is the external center of similitude of the circles C2 and C3. Hence, the external centers of similitude of three circles are collinear (1, paragraph 229, page 151). R and S are antihomologous points (4). The circle C3 touch the circles C1 and C2 in like manner (2, paragraph 408, page 189). The circles C1 and C2 constitute a coaxal system of the hyperbolic type (24 or 25) whose limiting points are the points U and V. The point U lies within the circle C1 and V lies within the circle C2. The line of centers, (2, paragraph 444, page 202), of the coaxal system is the line l. Further, let E, F be the common points of circles C1 and C2, respectively, with side BC of the triangle ABC and let K and L the common points of circles C1 and C2, respectively, with the segment AD. If the points M and N are the midpoints of the segments ED and KL, respectively, the line MN is the radical axis of the coaxal system (2, paragraph 427, b, page 195). Furthermore, the points E, F and T are collinear. Indeed, the two circles have an external common tangent and this tangent pass through the external center of similitude point T of the two circles (2, paragraph 396, a, page 185). E and F are antihomologous points (1, paragraph 30, corollary, page 21). If we draw the external circle of antisimilitude C4, whose center coincides with the external center of similitude T, of the circles C1 and C2, the circles C1, C2 and C4 are coaxal (3, paragraph 114, 1, page 227). A circle of antisimilitude of two circles is a circle with regard to which they are mutually inverse. Now consider the transformation by inversion I C4 (an inversion whose circle of inversion is C4). The points E and R are the images of F and S, respectively, and the circle C3 is anallagmatic (unchanged by inversion) (5). Also the point B is the image of the C. Hence the circle C3 is orthogonal to external circle of antisimilitude. Additionally, consider the circle C5 whose center G is midpoint of arc BC of circumcircle C3 of the triangle ABC which does not contain the vertex A and that passes through vertices B and C. The inversion I (C5) carries the circumcircle C3 into side line BC (6). This transformation carries the vertex A into point X foot of the internal bisector of the angle BAC. Indeed, if only the vertex A varies on arc BC (fixed) which does not contain the point G, the inversion I C5 carries the vertex A into corresponding foot of the internal bisector of the angle BAC. The circle C6 whose diameter is the segment AX (with center Y) is orthogonal to circle C5 (10, Theorem 44, item 2, page 91). The circles C1 and C3 form a coaxal system of tangent circles whose limiting point is the point R. As the segment BC is a chord of C3 and a tangent to the C1, the angle which the chord BC subtends at limiting point is bisected by line drawn from limiting point R to the point of contact E (7). Similarly, the circles C2 and C3 form a coaxal system of tangent circles whose limiting point is the point S and therefore the segment SF is the internal bisector of the triangle BSC. Hence, by the inversion I (C5), the points E and F are the images of the points R and S, respectively, and the circles C1 and C2 are anallagmatic with regard to C5 and thus, the circle C5 belong to the conjugate coaxal system determined by coaxal system (C1 and C2) (10, Theorem 60, 1, page 130). Consequently, the circle C5 pass by limiting points U and V and the point G lies on the radical axis of coaxal system (C1 and C2). The common tangents EF and KL to the two circles C1 and C2 subtend right angles at the limiting points U and V (3, paragraph 88, corollary 3, page 178). Therefore, the points E, F, U and V are concyclic. In this circle, C7, the diameter is the segment EF. Similarly, the points K, U, L and V are concyclic. In this other circle, C8, the diameter is the segment KL. The triangles EUF and KUL are right-angled triangles whose circumcenters are the points M and N, respectively. The circumcircles of the triangles EUF and KUL are the circles C7 and C8, respectively. The circles C5, C7 and C8 form a coaxal system of the elliptic type whose basic points (2, paragraph, 445, page 202) are the points U and V. It is easy to see that the triangles EUF and KUL are equiangular and inversely similar. Indeed, if the segment FL intersect the segment KE at point W, the angles DFL, DLF and KLW are equal. Further, the angles DEK and DKE are equal. Hence, the angles EWF and KWL are equal to 90 degrees and the point W lies on the circle C7 and also at circle C8. As the point W lies within the circle C1, it coincides with the limiting point U. It is not difficult to find the axis and the center of the Dilative Reflection (9). Thus the triangle KUL is obtained from triangle EUF by a dilative reflection with center U, axis UF and similarity coefficient k=KU/UE. Consequently, the points E, U and K are collinear (perpendicular to axis of similitude UF at U) as well as the points U, L and F (axis of similitude). The segment EK is the polar of the point D with regard to circle C1 (8, paragraph 245, page 153). As the point U lies on the polar of the point D, the point D lies on the polar of the point U (8, paragraph, 247, page 154). If drawn the line f (polar of the point U with regard to circle C1), it is perpendicular to line l, line of centers of the coaxal system (C1 and C2), at point V (11, theorem 14, page 27 or 12). The circle C9 whose diameter is the segment UD pass by limiting point U and V. The line f (polar of the point U) meets the circle C5 at points V inverse of U and J, then the triangle UVJ is a right-angled triangle at V and the points U, G and J are collinear. The point G is the midpoint of UJ. The circles C1 and C6 constitute a coaxal system, but the circles C1 and C6 are anallagmatic with regard to circle C5. Hence, the circle C5 belongs to conjugate coaxal system determined by circles C1 and C6 (10, Theorem 60, page 130). The circle C5 is the common orthogonal circle of three conjugate coaxal systems determined by circles C1, C2 and C6 taken by pairs. Therefore, the point G is at radical axis of the coaxal systems formed by circles C1, C2 and C6 taken by pairs. The point G is radical center of the circles C1, C2 and C6 and, more, the circle C5 is radical circle of the three circles. If the line z is the polar of the point U with regard to circle C6, the polars of the point U with respect to the circles of the coaxal system determined by circles C1 and C6 (line f and line z) are concurrent at point Z (10, theorem 61, page 131). Similarly, the polars of the point U with respect to the circles of the coaxal system determined by circles C2 and C6 (line f and line z) are concurrent at point Z (11, theorem 22, page 71-72). The polars of the point U with respect to the circles of the coaxal system determined by circles C1 and C2 coincide with the line f because the points U and V are inverse points with regard to all circles belong to the coaxal circles determined by circles C1 and C2. The midpoint of the segment ZU is at radical axis formed by circles C1 and C6 and at radical axis formed by circles C2 and C6 (11, theorem 22, page 71-72). Additionally, segment ZU is the diameter of the common orthogonal circle belong to the conjugate coaxal systems to coaxal systems (C1 and C6) and (C2 and C6). It is easy to see that in this case the point Z coincides with the point J. The points U and J are called Polar Conjugates (11, theorem 22, page 72). If we construct the circles C10 and C11, (2, paragraph 451, page 204), circles belong to coaxal system determined by circles C1 and C6 that passes through the points U and J, respectively, the line UJ is the external common tangent to circles C10 and C11 (11, theorem 22, page 72). Similarly, if, again, construct the circles C12 and C13, circles belong to coaxal system determined by circles C6 and C2 that passes through the points U and J, respectively, the line UJ is the external common tangent to circles C12 and C13. If two circles have external common tangents, these tangents pass through the external center of similitude of the two circles (2, paragraph 396, a, page 185). Hence, the line UJ passes through external center of similitude of the circles C10 and C11. Likewise, the line UJ passes through external center of similitude of the circles C12 and C13. The external center of similitude of the circles C10 and C11 is at line of centers of coaxal system formed by circles C6 and C1. Similarly, the external center of similitude of the circles C12 and C13 is at line of centers of coaxal system formed by circles C6 and C2. The lines of centers of the two coaxal systems intersect at point Y. The external circle of antisimilitude C14, whose center is the external center of similitude, of the circles C10 and C11 is coaxal with them (1, paragraphs 126 and 127, page 96). The inversion I (C14) carries the circle C10 into circle C11. Therefore, carries the point U into point J, but if the point J is the image of the point U, the circle C13 is the image, by same transformation I (C14), of the circle C12. If the inversion I (C14) carries the circle C12 into circle C13, the circles C14, C12 and C13 are coaxal. Hence the external centers of similitude of pairs of circles, (C10 and C11) and (C12 and C13) are identical and is the common point of the lines of centers of the two coaxal systems. Clearly, the common external center of similitude is the center of the circle C6, the point Y. Therefore, the circle C6 is the common external circle of the antisimilitude of pairs circles (C10 and C11) and (C12 and C13). The two coaxal systems cannot have more than one circle in common (2, paragraph 441, page 201). The circle C14 is identical with the circle C6. The inversion I (C6) carries the point U into point J. Hence, the points Y, U, G and J are collinear. Consequently, the points U and J are at diameter AX and the points A, Y, U, X, G and J are collinear. The points U and J are the incenter I and the excenter, respectively, of the triangle ABC. Therefore, the limiting point U of the coaxal system formed by circles C1 and C2 is incenter I of the triangle ABC. The limiting points are at line of centers of the coaxal system formed by the circles C1 and C2. Hence, the points P, Q and I are collinear. If we draw the other external common tangent and the internal common tangents to circles C1 and C2, there are four triangles which satisfy the given conditions of the problem. The point U is the incenter of the two triangles and the point V is the incenter of the two others triangles. It is important to observe that the circles C1, C2 and C4 form a hyperbolic coaxal system (limiting points U and V) and its conjugate elliptic coaxal system is formed by circles C5, C7, C8, C9 and C15 (orthogonal circle of the three circles C1, C2 and C3). The circle C15 pass by points limiting points of the circles C1, C2, and C3 taken by pairs. Hence, the points R, U (or incenter of two triangles), V (or incenter of two triangles) and S are concyclic.

>

> References

>

> (1) Roger Arthur Johnson, Advanced Euclidean Geometry (Modern Geometry) An Elementary Treatise on the Geometry of the Triangle and the Circle, Dover, 1960.

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> (3) William J. MClelland, A Treatise on the Geometry of the Circle and some Extensions to Conic Sections by the Method of Reciprocation, Macmillan and CO, 1891.

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>

> Sincerely,

>

> Deoclecio Gouveia Mota Junior.

>