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Forming an equilateral hexagon

Source: Tournament of Towns,Spring 2002, Senior A Level, P5

May 14, 2014
geometrytrigonometryrhombusgeometry proposed

Problem Statement

Let AA1,BB1,CC1AA_1,BB_1,CC_1 be the altitudes of acute ΔABC\Delta ABC. Let Oa,Ob,OcO_a,O_b,O_c be the incentres of ΔAB1C1,ΔBC1A1,ΔCA1B1\Delta AB_1C_1,\Delta BC_1A_1,\Delta CA_1B_1 respectively. Also let Ta,Tb,TcT_a,T_b,T_c be the points of tangency of the incircle of ΔABC\Delta ABC with BC,CA,ABBC,CA,AB respectively. Prove that TaOcTbOaTcObT_aO_cT_bO_aT_cO_b is an equilateral hexagon.
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