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Circumcircles and concurrence

Source: Philippine MO 2023/3

March 19, 2023
geometrycircumcirclePMO

Problem Statement

In △ABC\triangle ABC, AB>ACAB > AC. Point PP is on line BCBC such that APAP is tangent to its circumcircle. Let MM be the midpoint of ABAB, and suppose the circumcircle of △PMA\triangle PMA meets line ACAC again at NN. Point QQ is the reflection of PP with respect to the midpoint of segment BCBC. The line through BB parallel to QNQN meets PNPN at DD, and the line through PP parallel to DMDM meets the circumcircle of △PMB\triangle PMB again at EE. Show that the lines PMPM, BEBE, and ACAC are concurrent.
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