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point lies on midline, circles having AB,BH,CH,AC as diameters

Source: 2018 XXI All-Ukrainian Tournament of Young Mathematicians named after M. Y. Yadrenko, Qualifying p18

May 5, 2021
midlinegeometrycirclesUkrainian TYM

Problem Statement

In the acute triangle ABCABC, the altitude AHAH is drawn. Using segments AB,BH,CHAB,BH,CH and ACAC as diameters circles ω1,ω2,ω3\omega_1, \omega_2, \omega_3 and ω4\omega_4 are constructed respectively. Besides the point HH, the circles ω1\omega_1 and ω3\omega_3 intersect at the point P,P, and the circles ω2\omega_2 and ω4\omega_4 interext at point QQ. The lines BQBQ and CPCP intersect at point NN. Prove that this point lies on the midline of triangle ABCABC, which is parallel to BCBC.
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