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Perpendiculars from isogonal conjugates on angle-bisector

Source: Kvant Magazine No. 9 2019 M2577*

March 14, 2023
geometryisogonal conjugatesKvant

Problem Statement

Inside the acute-angled triangle ABCABC we take PP and QQ two isogonal conjugate points. The perpendicular lines on the interior angle-bisector of ∠BAC\angle BAC passing through PP and QQ intersect the segments ACAC and ABAB at the points Bp∈ACB_p\in AC, Bq∈ACB_q\in AC, Cp∈ABC_p\in AB and Cq∈ABC_q\in AB, respectively. Let WW be the midpoint of the arc BACBAC of the circle (ABC)(ABC). The line WPWP intersects the circle (ABC)(ABC) again at P1P_1 and the line WQWQ intersects the circle (ABC)(ABC) again at Q1Q_1. Prove that the points P1P_1, Q1Q_1, BpB_p, BqB_q, CpC_p and CqC_q lie on a circle.
Proposed by P. Bibikov
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