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another romanian concurrency

Source: IMAR 2015 p3

September 26, 2018
geometryconcurrencyconcurrentperpendicularantipode

Problem Statement

Let ABCABC be a triangle, let A1,B1,C1A_1, B_1, C_1 be the antipodes of the vertices A,B,CA, B, C, respectively, in the circle ABCABC, and let XX be a point in the plane ABCABC, collinear with no two vertices of the triangle ABCABC. The line through BB, perpendicular to the line XBXB, and the line through CC, perpendicular to the line XCXC, meet at A2A_2, the points B2B_2 and C2C_2 are defined similarly. Show that the lines A1A2,B1B2A_1A_2, B_1B_2 and C1C2C_1C_2 are concurrent.
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