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Many Concurrencies

Source: KöMaL A. 820

March 23, 2022
geometrykomal

Problem Statement

Let ABCABC be an arbitrary triangle. Let the excircle tangent to side aa be tangent to lines AB,BCAB,BC and CACA at points Ca,Aa,C_a,A_a, and Ba,B_a, respectively. Similarly, let the excircle tangent to side bb be tangent to lines AB,BC,AB,BC, and CACA at points Cb,Ab,C_b,A_b, and Bb,B_b, respectively. Finally, let the excircle tangent to side cc be tangent to lines AB,BC,AB,BC, and CACA at points Cc,Ac,C_c,A_c, and Bc,B_c, respectively. Let A′A' be the intersection of lines AbCbA_bC_b and AcBc.A_cB_c. Similarly, let B′B' be the intersection of lines BaCaB_aC_a and AcBc,A_cB_c, and let CC be the intersection of lines BaCaB_aC_a and AbCb.A_bC_b. Finally, let the incircle be tangent to sides a,b,a,b, and cc at points Ta,Tb,T_a,T_b, and Tc,T_c, respectively.
a) Prove that lines A′Aa,B′Bb,A'A_a,B'B_b, and C′CcC'C_c are concurrent.
b) Prove that lines A′Ta,B′Tb,A'T_a, B'T_b, and C′TcC'T_c are also concurrent, and their point of intersection is on the line defined by the orthocentre and the incentre of triangle ABC.ABC.
Proposed by Viktor Csaplár, Bátorkeszi and Dániel Hegedűs, Gyöngyös
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