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Perpendicular lines

Source: Czech and Slovak Olympiad 2019, National Round, Problem 4

June 7, 2019
geometryexcircleTriangleperpendicular linesnational olympiad

Problem Statement

Let be ABCABC an acute-angled triangle. Consider point PP lying on the opposite ray to the ray BCBC such that ∣AB∣=∣BP∣|AB|=|BP|. Similarly, consider point QQ on the opposite ray to the ray CBCB such that ∣AC∣=∣CQ∣|AC|=|CQ|. Denote JJ the excenter of ABCABC with respect to AA and D,ED,E tangent points of this excircle with the lines ABAB and ACAC, respectively. Suppose that the opposite rays to DPDP and EQEQ intersect in F≠JF\neq J. Prove that AF⊥FJAF\perp FJ.
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