On Mixtilinear Incircles and Excircles

A mixtilinear incircle (respectively excircle) of a triangle is tangent to two sides and to the circumcircle internally (respectively externally). We study the configuration of the three mixtilinear incircles (respectively excircles). In particular, we give easy constructions of the circle (apart from the circumcircle) tangent the three mixtilinear incircles (respectively excircles). We also obtain a number of interesting triangle centers on the line joining the circumcenter and the incenter of a triangle. 1. Preliminaries In this paper we study two triads of circles associated with a triangle, the mixtilinear incircles and the mixtilinear excircles. For an introduction to these circles, see [4] and §§2, 3 below. In this section we collect some important basic results used in this paper. Proposition 1 (d’Alembert’s Theorem [1]). Let O1(r1), O2(r2), O3(r3) be three circles with distinct centers. According as e = +1 or −1, denote by A1e, A2e, A3e respectively the insimilicenters or exsimilicenters of the pairs of circles ((O2), (O3)), ((O3), (O1)), and ((O1), (O2)). For ei = ±1, i = 1, 2, 3, the points A1e1 , A2e2 and A3e3 are collinear if and only if e1e2e3 = −1. See Figure 1. The insimilicenter and exsimilicenter of two circles are respectively their internal and external centers of similitude. In terms of one-dimensional barycentric coordinates, these are the points ins(O1(r1), O2(r2)) = r2 ·O1 + r1 ·O2 r1 + r2 , (1) exs(O1(r1), O2(r2)) = −r2 ·O1 + r1 ·O2 r1 − r2 . (2) Proposition 2. LetO1(r1), O2(r2), O3(r3) be three circles with noncollinears centers. For e = ±1, let Oe(re) be the Apollonian circle tangent to the three circles, all externally or internally according as e = +1 or −1. Then the Monge line containing the three exsimilicenters exs(O2(r2), O3(r3)), exs(O3(r3), O1(r1)), and exs(O1(r1), O2(r2)) is the radical axis of the Apollonian circles (O+) and (O−). See Figure 1. Publication Date: January 18, 2006. Communicating Editor: Paul Yiu. The authors thank Professor Yiu for his contribution to the last section of this paper. 2 K. L. Nguyen and J. C. Salazar