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Two lines intersent on a circle

Source: Israeli Olympic Revenge 2021, Problem 3

August 29, 2021
geometrycircumcircle

Problem Statement

Let ABCABC be a triangle. A point PP is chosen inside △ABC\triangle ABC such that ∠BPC+∠BAC=180∘\angle BPC+\angle BAC=180^{\circ}. The lines AP,BP,CPAP,BP,CP intersect BC,CA,ABBC,CA,AB at PA,PB,PCP_A,P_B,P_C respectively. Let XAX_A be the second intersection of the circumcircles of △ABC\triangle ABC and △APBPC\triangle AP_BP_C . Similarly define XB,XCX_B,X_C. Let B′B' be the intersection of lines AXA,CXCAX_A,CX_C, and let C′C' be the intersection of lines AXA,BXBAX_A,BX_B. Prove that lines BB′BB' and CC′CC' intersect on the circumcircle of △APBPC\triangle AP_BP_C.
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