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Triangle Centers Associated with the Malfatti Circles Forum Geometricorum b Volume 3 (2003) 83–93. bbb FORUM GEOM ISSN 1534-1178 Triangle Centers Associated with the Malfatti Circles Milorad R. Stevanovi´c Abstract. Various formulae for the radii of the Malfatti circles of a triangle are presented. We also express the radii of the excircles in terms of the radii of the Malfatti circles, and give the coordinates of some interesting triangle centers associated with the Malfatti circles. 1. The radii of the Malfatti circles The Malfatti circles of a triangle are the three circles inside the triangle, mutually tangent to each other, and each tangent to two sides of the triangle. See Figure 1. Given a triangle ABC, let a, b, c denote the lengths of the sides BC, CA, AB, s the semiperimeter, I the incenter, and r its inradius. The radii of the Malfatti circles of triangle ABC are given by C X3 r3 Y3 r3 O3 C3 X2 C2 r I 2 C Y1 1 O2 r1 O1 r2 r1 A Z1 Z2 B Figure 1 r r = (s − r − (IB + IC − IA)) , 1 2(s − a) r r = (s − r − (IC + IA − IB)) , 2 2(s − b) (1) r r = (s − r − (IA + IB − IC)) . 3 2(s − c) Publication Date: March 24, 2003. Communicating Editor: Paul Yiu. The author is grateful to the referee and the editor for their valuable comments and helps. 84 M. R. Stevanovi´c According to F.G.-M. [1, p.729], these results were given by Malfatti himself, and were published in [7] after his death. See also [6]. Another set of formulae give the same radii in terms of a, b, c and r: (IB + r − (s − b))(IC + r − (s − c)) r = , 1 2(IA + r − (s − a)) (IC + r − (s − c))(IA + r − (s − a)) r = , 2 2(IB + r − (s − b)) (2) (IA + r − (s − a))(IB + r − (s − b)) r = . 3 2(IC + r − (s − c)) These easily follow from (1) and the following formulae that express the radii r1, r2, r3 in terms of r and trigonometric functions: 1+tanB 1+tanC r r = 4 4 · , 1 1+tanA 2 4 C A 1+tan 4 1+tan 4 r r2 = · , (3) 1+tanB 2 4 1+tanA 1+tanB r r = 4 4 · . 3 C 2 1+tan 4 These can be found in [10]. They can be used to obtain the following formula which is given in [2, pp.103–106]. See also [12]. 2 1 1 1 1 1 1 = √ + √ + √ − + + . (4) r r1r2 r2r3 r1r2 r1r2 r2r3 r3r1 2. Exradii in terms of Malfatti radii Antreas P. Hatzipolakis [3] asked for the exradii ra, rb, rc of triangle ABC in terms of the Malfatti radii r1, r2, r3 and the inradius r. Proposition 1. 2 − √ 1 r r2r3 ra − r1 = , 2 − √ 1 2 − √ 1 r r3r1 r r1r2 2 − √ 1 r r3r1 rb − r2 = , (5) 2 − √ 1 2 − √ 1 r r1r2 r r2r3 2 − √ 1 r r1r2 rc − r3 = . 2 − √ 1 2 − √ 1 r r2r3 r r3r1 Triangle centers associated with the Malfatti circles 85 Proof. For convenience we write A B C t := tan ,t:= tan ,t:= tan . 1 4 2 4 3 4 A B C Note that from tan 4 + 4 + 4 =1,wehave 1 − t1 − t2 − t3 − t1t2 − t2t3 − t3t1 + t1t2t3 =0. (6) From (3) we obtain 2 1 t 2 − √ = 1 · , r r2r3 1+t1 r 2 1 t2 2 − √ = · , (7) r r3r1 1+t2 r 2 1 t 2 − √ = 3 · . r r1r2 1+t3 r For the exradius ra,wehave 2 2 s B C (1 − t2)(1 − t3) ra = · r =cot cot · r = · r. s − a 2 2 4t2t3 It follows that r (1 − t2)(1 − t3) 1 ra − r1 =(1 + t2)(1 + t3) · − 2 2t2t3 1+t1 r (1 + t1)(1 − t2)(1 − t3) − 2t2t3 =(1 + t2)(1 + t3) · · 2 2t2t3(1 + t1) r 2t1 =(1 + t2)(1 + t3) · · (from (6)) 2 2t2t3(1 + t1) t 1+t 1+t r = 1 · 2 · 3 · . 1+t1 t2 t3 2 Now the result follows from (7). Note that with the help of (4), the exradii ra, rb, rc can be explicitly written in terms of the Malfatti radii r1, r2, r3. We present another formula useful in the next sections in the organization of coordinates of triangle centers. Proposition 2. 1 1 a (1 + cos B )(1 + cos C ) − = · 2 2 . r r rs A 1 a 1+cos 2 3. Triangle centers associated with the Malfatti circles Let A be the point of tangency of the Malfatti circles C2 and C3. Similarly define B and C. It is known ([4, p.97]) that triangle ABC is perspective with ABC at the first Ajima-Malfatti point X179. See Figure 3. We work out the details here and construct a few more triangle centers associated with the Malfatti circles. In particular, we find two new triangle centers P+ and P− which divide the incenter I and the first Ajima-Malfatti point harmonically. 86 M. R. Stevanovi´c 3.1. The centers of the Malfatti circles. We begin with the coordinates of the cen- ters of the Malfatti circles. Since O1 divides the segment AIa in the ratio AO1 : O1Ia = r1 : ra − r1, O1 1 1 1 rs = − A + · IA ra = we have r1 r1 ra ra . With s−a we rewrite the absolute barycentric coordinates of O1, along with those of O2 and O3, as follows. O 1 1 s − a 1 = − A + · I , r r r rs a 1 1 a O 1 1 s − b 2 = − B + · I , r r r rs b (8) 2 2 b O3 1 1 s − c = − C + · Ic. r3 r3 rc rs From these expressions we have, in homogeneous barycentric coordinates, 1 1 O = 2rs − − a : b : c , 1 r r 1 a 1 1 O = a :2rs − − b : c , 2 r r 2 b 1 1 O3 = a : b :2rs − − c . r3 rc C X3 Y3 O3 A X2 B P− O2 Y1 O1 C A Z1 Z2 B Figure 2 Triangle centers associated with the Malfatti circles 87 3.2. The triangle center P−. It is clear that O1O2O3 is perspective with ABC at the incenter (a : b : c). However, it also follows that if we consider A = BO3 ∩ CO2,B= CO1 ∩ AO3,C= AO2 ∩ BO1, then triangle ABC is perspective with ABC at 1 1 1 1 1 1 P = 2rs − − a :2rs − − b :2rs − − c − r r r r r r 1 a 2 b 3 c 1 1 a 1 1 b 1 1 c = − − : − − : − − r r 2rs r r 2rs r r 2rs 1 a 2 b 3 c (1 + cos B )(1 + cos C ) 1 = a 2 2 − : ···: ··· A 2 (9) 1+cos 2 by Proposition 2. See Figure 2. Remark. The point P− appears in [5] as the first Malfatti-Rabinowitz point X1142. 3.3. The first Ajima-Malfatti point. For the points of tangency of the Malfatti cir- cles, note that A divides O2O3 in the ratio O2A : A O3 = r2 : r3. We have, in absolute barycentric coordinates, 1 1 O O a 1 1 1 1 + A = 2 + 3 = · A + − B + − C; r2 r3 r2 r3 rs r2 rb r3 rc similarly for B and C. In homogeneous coordinates, a 1 1 1 1 A = : − : − , rs r r r r 2 b 3 c 1 1 b 1 1 B = − : : − , r r rs r r (10) 1 a 3 c 1 1 1 1 c C = − : − : . r1 ra r2 rb rs From these, it is clear that ABC is perspective with ABC at 1 1 1 1 1 1 P = − : − : − r r r r r r 1 a 2 b 3 c a(1 + cos B )(1 + cos C ) b(1 + cos C )(1 + cos A ) = 2 2 : 2 2 1+cosA 1+cosB 2 2 c(1 + cos A )(1 + cos B ) : 2 2 1+cosC 2 a b c = : : A 2 B 2 C 2 (11) 1+cos 2 1+cos 2 1+cos 2 88 M. R. Stevanovi´c by Proposition 2. The point P appears as X179 in [4, p.97], with trilinear coordi- nates A B C sec4 :sec4 :sec4 4 4 4 computed by Peter Yff, and is named the first Ajima-Malfatti point. See Figure 3. C O3 B A P O2 C O1 A B Figure 3 3.4. The triangle center P+. Note that the circle through A , B , C is orthogonal to the Malfatti circles. It is the radical circle of the Malfatti circles, and is the incircle of O1O2O3. The lines O1A , O2B , O3C are concurrent at the Gergonne point of triangle O1O2O3. See Figure 4. As such, this is the point P+ given by 1 1 1 O O O + + P = 1 + 2 + 3 r r r + r r r 1 2 3 1 2 3 1 1 I 1 1 I 1 1 I = − A + a + − B + b + − C + c r r r r r r r r r 1 a a 2 b b 3 c c 1 1 1 1 1 1 I = − A + − B + − C + r r r r r r r 1 a 2 b 3 c 1 1 1 1 1 1 1 = − A + − B + − C + (aA + bB + cC) r r r r r r 2rs 1 a 2 b 3 c 1 1 a 1 1 b 1 1 c = − + A + − + B + − + C. r1 ra 2rs r2 rb 2rs r3 rc 2rs Triangle centers associated with the Malfatti circles 89 It follows that in homogeneous coordinates, 1 1 a 1 1 b 1 1 c P = − + : − + : − + + r r 2rs r r 2rs r r 2rs 1 a 2 b 3 c (1 + cos B )(1 + cos C ) 1 = a 2 2 + : ···: ··· A 2 (12) 1+cos 2 by Proposition 2.
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