Problem

Source: TSTST 2021/6

Tags: USA TSTST, geometry



Triangles ABC and DEF share circumcircle Ω and incircle ω so that points A,F,B,D,C, and E occur in this order along Ω. Let ΔA be the triangle formed by lines AB,AC, and EF, and define triangles ΔB,ΔC,,ΔF similarly. Furthermore, let ΩA and ωA be the circumcircle and incircle of triangle ΔA, respectively, and define circles ΩB,ωB,,ΩF,ωF similarly. (a) Prove that the two common external tangents to circles ΩA and ΩD and the two common external tangents to ωA and ωD are either concurrent or pairwise parallel. (b) Suppose that these four lines meet at point TA, and define points TB and TC similarly. Prove that points TA,TB, and TC are collinear. Nikolai Beluhov