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concyclic wanted, 2 circles related

Source: 1st Mathematics Regional Olympiad of Mexico Northwest 2018 P3

September 6, 2022
geometryConcyclic

Problem Statement

Let ABCABC be an acute triangle orthocenter angle HH. Let ω1\omega_1 be the circle tangent to BCBC at BB and passing through HH and ω2\omega_2 the circle tangent to BCBC at CC and passing through through HH. A line ℓ\ell passing through HH intersects the circles ω1\omega_1 and ω2\omega_2 at points DD and EE, respectively (with DD and EE other than HH). Lines BDBD and CECE intersect at FF, the lines ℓ\ell and AFAF intersect at XX and the circles ω1\omega_1 and ω2\omega_2 intersect at the points PP and HH. Prove that the points A,H,PA, H, P and XX are still on the same circle.
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