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some angle chasing

Source: III Caucasus Mathematical Olympiad

March 17, 2018
geometry

Problem Statement

In an acute-angled triangle ABCABC, the altitudes from A,B,CA,B,C meet the sides of ABCABC at A1A_1, B1B_1, C1C_1, and meet the circumcircle of ABCABC at A2A_2, B2B_2, C2C_2, respectively. Line A1C1A_1 C_1 intersects the circumcircles of triangles AC1C2AC_1 C_2 and CA1A2CA_1 A_2 at points PP and QQ (Q≠A1Q\neq A_1, P≠C1P\neq C_1). Prove that the circle PQB1PQB_1 touches the line ACAC.
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