On Emelyanov s Circle Theorem

Transcription

1 Journal for Geometry and Graphics Volume 9 005, No., On Emelyanov s ircle Theorem Paul Yiu Department of Mathematical Sciences, Florida Atlantic University Boca Raton, Florida, 3343, USA yiu@fau.edu Abstract. Given a triangle and a point T, let Γ + T be the triad of circles each tangent to the circumcircle and to a side line at the trace of T. Assuming T an interior point and each circle tangent the circumcircle internally, Lev Emelyanov has shown that the circle tangent to each of these circles is also tangent to the incircle. In this paper, we study this configuration in further details and without the restriction to interior points. We identify the point of the tangency with the incircle, and derive some interesting loci related this configuration. Key words: Emelyanov circle, homogeneous barycentric coordinates, infinite point, nine-point circle, Feuerbach point MS 000: 5M04. Introduction Lev Emelyanov has established by synthetic method the following remarkable theorem: Theorem Emelyanov [] Let A, B, be the traces of an interior point T on the side lines of triangle AB. onstruct three circles Γ a +, Γ b + and Γ c + outside the triangle which are tangent to the sides at A, B,, respectively, and also tangent to the circumcircle of AB. The circle tangent externally to these three circles is also tangent to the incircle of triangle AB see Fig.. In the present paper we study this configuration from the viewpoint of geometric constructions facilitated by a computer software which allows the definitions of macros to perform the constructions efficiently and elegantly. To achieve this we analyze the underlying geometry by using the language and basic results of triangle geometry. Specifically, we make use of barycentric coordinates and their homogenization see, for example, [3, 4]. We make use of standard notations: a, b, c denote the lengths of the sides of triangle AB, opposite to A, B, respectively; s stands for the semiperimeter a + b + c. Apart from the most basic ISSN /$.50 c 005 Heldermann Verlag

2 56 Paul Yiu: On Emelyanov s circle theorem A Γ b + Γ c + Z + Z I R O T Y B Y + B X A Γ a + X + Figure : Emelyanov circle Table : Triangle centers and their barycentric coordinates centroid G : : incenter I a : b : c circumcenter O a b + c a : b c + a b : c a + b c orthocenter H : : b + c a c + a b a + b c symmedian point K a : b : c Gergonne point G e s a : s b : s c Nagel point N s a : s b : s c Mittenpunkt M as a : bs b : cs c Feuerbach point F s ab c : s bc a : s ca b triangle centers listed below with their homogeneous barycentric coordinates, we adopt the labeling of triangle centers in []. Dynamic sketches illustrating results in this paper can be found in the author s website

3 Paul Yiu: On Emelyanov s circle theorem 57 The triad of circles Γ a +, Γ b + and Γ c + can be easily constructed with the help of the following simple lemma. A + A + O P A X + Q X l X O Q X + A P l Figure : onstruction of a circle tangent to l at P and to Lemma Given a line l and a circle, let A + and A be the endpoints of the diameter of perpendicular to l. a If a circle touches both l and, the line joining the points of tangency passes through one of the points A ±. b If two circles are tangent to the line l at the same point, the line joining their points of tangency with passes through the pole of l with respect to see Figs. A and B. omputations in this paper were performed with the aids of a computer algebra system. Theorem 4 below characterizes the points of the tangency with the circumcircle. Here, we make use of the notion of the barycentric product of two points. Given two points P = u : v : w and P = u : v : w in homogeneous barycentric coordinates, the barycentric product P P is the point with coordinates u u : v v : w w. Here is a simple construction [3]: Let X, X be the traces of P and P on the line B, distinct from the vertices B,. For i =,, complete the parallelograms AK i X i H i with K i on AB and H i on A. The line joining the intersections of BH and K, and of BH and K, passes through the vertex A, and intersects B at a point X with homogeneous coordinates 0 : v v : w w. This is the trace of the point P P on B. The traces of the same point on the lines A and AB can be similarly constructed. From any two of these traces, the barycentric product P P can be determined see Fig. 3.. The triads of circles Γ ± T onsider a triangle AB with circumcircle. We label A ± the endpoints of the diameter perpendicular to the sideline B, A + being the one on the same side of B as A, A the antipodal point.

4 58 Paul Yiu: On Emelyanov s circle theorem A K H K P P H B X X X Figure 3: onstruction of trace X of P P on B Lemma 3 For ε = ±, similarly, for B ± and ±. A ε = a : bb εc : cc εb ; Proof: The infinite point of lines perpendicular to B can be taken as a, a + b c, c + a b. We seek t such that t0,, + a, a + b c, c + a b lies on the circumcircle a yz + b zx + c xy = 0. This gives t = b + εbc + c a for ε = ±. Thus, the perpendicular bisector of B intersects the circumcircle A ε given above. Given a point T, the triad Γ + T consists of the three circles Γ a +, Γ b +, Γ c +, each tangent to the circumcircle and the sidelines at the traces of T, the lines joining the points of tangency passing through A +, B +, +, respectively. This is the triad considered in [] when T is an interior point of the triangle. We do not impose this restriction here. Similarly, we also consider the triad Γ T, consisting of the circles a, b, c. Theorem 4 a The points of tangency of the circumcircle with the circles in the triad Γ + T are the vertices of the circumcevian triangle of the barycentric product P = I T, where I is the incenter of triangle AB see Fig. 4. b Let the trilinear polar of P intersect B, A, AB at X, Y, Z respectively. The points of tangency of the circumcircle with the circles in the triad Γ T are the second intersections of AX, BY, Z with the circumcircle. c The line joining the points of tangency of the circumcircle with the corresponding circles in the two triads passes through the corresponding vertex of the tangential triangle see Fig. 5.

5 Paul Yiu: On Emelyanov s circle theorem 59 A Γ b + Γ c + O P T B B A Γ a + Figure 4: The triad Γ + T and P = I T Proof: Let T be a point with homogeneous barycentric coordinates u : v : w. a For ε = ±, the line joining A ε to 0 : v : w intersects the circumcircle again at a point with coordinates x, y, z = a, bb εc, cc εb + t0, v, w for some nonzero t satisfying. This gives t = bcv + w, and vw X ε = a vw : bvbw + εcv : cwcv + εbw. a Similarly, we obtain analogous expressions for Y ε and Z ε. From this, X + = vw bw + cv : bv : cw is on the line joining I T = au : bv : cw to A; similarly for Y + and Z +. It follows that X + Y + Z + is the circumcevian triangle of I T. a b On the other hand, the line joining X = vw bw cv : bv : cw to A intersects B at 0 : bv : cw; similarly for Y and Z. These three points lie on the line x au + y bv + z cw = 0, the trilinear polar of I T. c follows from Lemma b. Remark. The three lines in Theorem 4c above are concurrent at a point P = a b v + c w a u : b c w + a u b : c a v u + b v c. w

6 60 Paul Yiu: On Emelyanov s circle theorem B A Γ c Γ a Γ b + B Γ c + P O T B A B Γ b Γ a + A Figure 5: The triads Γ + T and Γ T Here are two simple examples. T P P G I O I K K 3. Equations of the circles in Γ + T Since the points of tangency of the circles in Γ + T with the circumcircle are known, the centers of the circles can be easily located: if the circle Γ a + touches the circumcircle at X +, then its center is the intersection of the lines OX + and the perpendicular to B at A ; similarly for the other two circles Γ b + and Γ c +. Proposition 5 The centers of the circles in the triad Γ + T are the points O = a vw : vss av + bb + cw : wss aw + cb + cv, O = uss bu + ac + aw : b wu : wss bw + cc + au, O 3 = uss cu + aa + bv : vss cv + ba + bu : c uv.

7 Paul Yiu: On Emelyanov s circle theorem 6 Proposition 6 These centers are collinear if and only if T lies on the union of the following curves: i the circumconic ii the curve m : E m : ayz + bzx + cxy = 0 s as bs c + abc xyzx + y + z + as bs cy z = 0. cyclic m A E m M B Figure 6: The curves E m and m Remarks. The circumconic E m is centered at the Mittenpunkt M = ab + c a : bc + a b : ca + b c. It is an ellipse since it does not contain any interior or infinite point. The curve m is the I-isoconjugate 3 of the circle a + b + c a bc caa yz + b zx + c xy + + x + y + z cyclic bcc + a ba + b cx = 0, with center at the Mittenpunkt M and radius s RR + r see Fig. 6. 4R + r The Mittenpunkt M is the intersection of the three lines each joining the midpoint of a side to the center of the excircle on that side. 3 a The I-isoconjugate of a point P = x : y : z is the point with coordinates x : b y : c z. It can be constructed as the barycentric product of the incenter and the isotomic conjugate of P.

8 6 Paul Yiu: On Emelyanov s circle theorem Proposition 7 The equations of the circles in the triad Γ + T are v + w a yz + b zx + c xy = x + y + z cv + bw x + a w y + a v z, w + u a yz + b zx + c xy = x + y + z b w x + aw + cu y + b u z, u + v a yz + b zx + c xy = x + y + zc v x + c u y + bu + av z. Proof: Since Γ a + is tangent to the circumcircle at X + = a vw : bvcv +bw : cwbw +cv, and the tangent of the circumcircle at this point is the equation of Γ a + is cv + bw x + a w y + a v z = 0, ka yz + b zx + c xy = x + y + z cv + bw x + a w y + a v z for a choice of k such that with x = 0, the equation has only one root. It is routine to verify that this is k = v + w. This gives the first of the equations above. The remaining two equations of Γ b + and Γ c + are easily obtained by cyclically permuting a, u, x, b, v, y, and c, w, z. 4. The radical center Q T and the Emelyanov circle T The radical center of the circles in the triad Γ + T can be determined by a formula given, for example, in [4, 7.3.]. Proposition 8 If T = u : v : w, the radical center of the triad Γ + T is the point 4 Q T = a as auvwu + v + w + bcu v + w a av w + bw u + cu v : :. orollary 9 The radical center Q T is a point on the circumcircle if and only if T is an infinite point or lies on the ellipse E m, an infinite point if T lies on the curve m. For T = G, this radical center is the point X 00 = aa ab + c bc : :, which divides the centroid G and X 55, the internal center of similitude of the circumcircle and incircle, in the ratio GX 00 : X 00 X 55 = R + r : 3R. The circle T Emelyanov established in [] is the one for which each of the circles in the triad Γ + T is oppositely tangent to both T and the circumcircle. This circle can be easily constructed since the radical center Q T is known. If the line joining Q T to X + respectively Y +, Z + intersects the circle Γ a + respectively Γ b +, Γ c + again at X respectively Y, Z, then the circle through X, Y and Z is the one Emelyanov constructed see Fig.. 4 The second and third components are obtained from the first by cyclically permuting a, b, c and u, v, w.

9 Paul Yiu: On Emelyanov s circle theorem 63 Proposition 0 The equation of the Emelyanov circle T is u + v + wa yz + b zx + c f xy x + y + z u x + g v y + h w z = 0, where where f = fu, v, w = kvw + s aavw + bwu + cuv, g = gu, v, w = kwu + s bavw + bwu + cuv, 3 h = hu, v, w = kuv + s cavw + bwu + cuv, 4 k = s as b + s bs c + s cs a. 5 Examples For the Gergonne point G e, the Emelyanov circle G e clearly coincides with the incircle. For the centroid G, the Emelyanov circle G has equation a yz + b zx + c xy x + y + z cyclica + ab + c 3b + bc + 3c x = 0. This has center 5 G = a 4 a 3 b + c a b bc + c + ab + cb c + b c : :. This point divides the segment between the incenter and the nine-point center in the ratio :. 3 From the equation of the circle, it is clear that T degenerates into a line if and only if T is an infinite point. 5. Point of tangency of the Emelyanov circle and the incircle The main result of [] states that the circle T touches the incircle of the reference triangle. Here we identify the point of tangency explicitly. Note that every point on the incircle has coordinates x s a : y s b : z s c for an infinite point x : y : z. Theorem The circle T is tangent to the incircle at the point F T = s a s bv : : s cw Note that F T corresponds to the infinite point of the line x s au + y s bv + z s cw = 0. Here is an alternative construction of the point F T The center of a circle with given equation can be determined by a formula given in, for example, [4, 0.7.].

10 64 Paul Yiu: On Emelyanov s circle theorem Proposition Let DEF be the intouch triangle of triangle AB, i.e., the cevian triangle of the Gergonne point. Given a point T = u : v : w with cevian triangle A B, let X = B EF, Y = A F D, Z = A B DE. The triangles XY Z and DEF are perspective at F T with coordinates given in 6 see Fig. 7. We note an immediate corollary of Theorem. orollary 3 The Emelyanov circle G touches the incircle at the Feuerbach point F = X = s ab c : s bc a : s ca b, the point of tangency of the incircle with the nine-point circle. Examples: 4 If T is an infinite point, the Emelyanov circle degenerates into a line tangent to the incircle. Now, a typical infinite point has coordinates b ca + t : c ab + t : a bc + t. The corresponding Emelyanov circle degenerates into a line tangent to the incircle at the point s ab c a + t aab + c b + c + ta ab + c b c : :. Z A Y F T X E F I T B B D A Figure 7: onstruction of F T

11 Paul Yiu: On Emelyanov s circle theorem 65 5 If T is a point on the curve m, we have seen in Proposition 6 and orollary 9 that the centers of the circles in Γ + T are collinear and the radical center Q T is an infinite point. If l is the line containing the centers, and if the circles in Γ + T touch the circumcircle at X, Y, Z respectively, then the Emelyanov circle contains the reflections of X, Y, Z in l. It is therefore the reflection of the circumcircle in l. 6. The locus of T for which T is tangent to the incircle at a specific point Theorem 4 Let T be a point other than the Gergonne point. The locus of T for which T touches the incircle at the same point as T does is the circum-hyperbola through T and the Gergonne point. Proof: By the remark following Theorem, for T = u : v : w and T = x : y : z, the circles T and T touch the incircle at the same point if and only if s bv s cw s ax + s cw s au s by + s au s bv s cz = 0. This defines a circumconic, which clearly contains T and the Gergonne point. circumconic contains an interior point the Gergonne point, it is a hyperbola. Since the Taking T to be the centroid, we obtain the following corollary. orollary 5 The locus of T for which T touches the incircle at the Feuerbach point is the circum-hyperbola through the centroid and the Gergonne point. Remark. This circumconic has equation b c x + c a y + a b z X 086 = b c : c a : a b = 0. Its center is the point on the Steiner in-ellipse. The tangent to the ellipse at this point is also tangent to the nine-point circle at the Feuerbach point. p More generally, for a given point Q = s a : q s b : r on the incircle, corresponding s c to an infinite point p : q : r, the locus of T for which the Emelyanov circle T touches the incircle at Q is the circum-hyperbola HQ: p s ax + q s by + r s cz = 0. To construct the hyperbola, we need only locate one more point. This can be easily found with the help of Proposition. Let the line DQ intersect A and AB at Y 0 and Z 0 respectively. Then T 0 = BY 0 Z 0 is a point on the locus. The hyperbola HQ is a rectangular hyperbola if it contains the orthocenter. This is the Feuerbach hyperbola with center at the Feuerbach point. For T on this hyperbola, the point of tangency of the Emelyanov circle T with the incircle is the point X 30 = a b c s a 3 : :.

12 66 Paul Yiu: On Emelyanov s circle theorem 7. The involution τ Theorem 6 Let fu, v, w, gu, v, w, and hu, v, w be the functions given in. mapping τu : v : w = fu, v, w : gu, v, w : hu, v, w is an involution such that the Emelyanov circles T and τt are identical. Proof: Let φt = τt. 6 It is enough to show that φ is an involution. If T = u : v : w, it is easy to check that φu : v : w = ku + s aau + bv + cw : kv + s bau + bv + cw 7 : kw + s cau + bv + cw, where k is given in 5. The mapping φ is an involution. In fact, it N denotes the Nagel point, and P the intersection of the lines NT and ax + by + cz = 0, then φt is the harmonic conjugate of T with respect to N and P. Since τt = φt, τ is also an involution. If T is the centroid, τt is the point a + ab + c 3b + bc + 3c : :. 7.. The fixed points of τ The fixed points of τ are the isotomic conjugates of the fixed points of φ. From 7, it is clear that the fixed points of φ are the Nagel point s a : s b : s c and the points on the line ax + by + cz = 0. Therefore, the fixed points of τ are the Gergonne point and points on the circum-ellipse E m. A typical point on E m is T = a b ca + t : b c ab + t : c. a bc + t For such a point T, the circles in the triad Γ + T are all tangent to the circumcircle at a P T = b : c :. b ca + t c ab + t a bc + t Every circle tangent to the circumcircle at this point is clearly tangent to the three circles of the triad Γ + T. There are two circles tangent to the incircle and also to the circumcircle at P T. One of them is the Emelyanov circle T. This touches the incircle at the point a s abca ab + c + b + c +ta b + c ab bc + c + b + cb c : :. In particular, the point of tangency is the Feuerbach if 7 T = X 673 = ab + c b + c : bc + a c + a : ca + b a + b The circles T and those in the triad Γ + T are tangent to the circumcircle at X Here, stands for isotomic conjugation. 7 X 673 can be constructed as the intersection of the lines joining the centroid to the Feuerbach point, and the Gergonne point to the symmedian point., The

13 Paul Yiu: On Emelyanov s circle theorem 67 Acknowledgment The author thanks the referees for valuable comments leading to improvement of this paper. References [] L. Emelyanov: A Feuerbach type theorem on six circles. Forum Geom., []. Kimberling: Encyclopedia of Triangle enters. available at evansville. edu/ck6/encyclopedia/et.html. [3] P. Yiu: The uses of homogeneous barycentric coordinates in plane euclidean geometry. Int. J. Math. Educ. Sci. Technol. 3, [4] P. Yiu: Introduction to the Geometry of the Triangle. Florida Atlantic University lecture notes, 00, available at Received April 5, 005; final form December 5, 005