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Locus geo, nicely generalizing the nine-point circle configuration

Source: Bulgaria MO Regional round 2024, 10.2

February 13, 2024
geometry

Problem Statement

Given are two fixed lines that meet at a point OO and form an acute angle with measure α\alpha. Let PP be a fixed point, internal for the angle. The points M,NM, N vary on the two lines (one point on each line) such that ∠MPN=180∘−α\angle MPN=180^{\circ}-\alpha and PP is internal for △MON\triangle MON. Show that the foot of the perpendicular from PP to MNMN lies on a fixed circle.
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