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IMOC 2018 G2 (collinearity given + wanted, 4 tangent circles)

Source: https://artofproblemsolving.com/community/c6h1740825p11314688

March 22, 2020
tangent circlescircumcirclegeometrycollinearcircles

Problem Statement

Given △ABC\vartriangle ABC with circumcircle Ω\Omega. Assume ωa,ωb,ωc\omega_a, \omega_b, \omega_c are circles which tangent internally to Ω\Omega at Ta,Tb,TcT_a,T_b, T_c and tangent to BC,CA,ABBC,CA,AB at Pa,Pb,PcP_a, P_b, P_c, respectively. If ATa,BTb,CTcAT_a,BT_b,CT_c are collinear, prove that APa,BPb,CPcAP_a,BP_b,CP_c are collinear.
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