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Three lines are concurrent on $OH$

Source: Israeli Olympic Revenge 2018, Problem 3

August 28, 2021
geometrycircumcircleEulerperpendicular bisector

Problem Statement

Let ABCABC be a triangle with circumcircle ω\omega and circumcenter OO. The tangent line to from AA to ω\omega intersects BCBC at KK. The tangent line to from BB to ω\omega intersects ACAC at LL. Let M,NM,N be the midpoints of AK,BLAK,BL respectively. The line MNMN is named by α\alpha. The feet of perpendicular from A,B,CA,B,C to the edges of △ABC\triangle ABC are named by D,E,FD,E,F respectively. The perpendicular bisectors of EF,DF,DEEF,DF,DE intersect α\alpha at X,Y,ZX,Y,Z respectively. Let AD,BE,CFAD,BE,CF intersect ω\omega again at D′,E′,F′D',E',F' respectively. If HH is the orthocenter of ABCABC, prove that the lines XD′,YE′,ZF′,OHXD',YE',ZF',OH are concurrent.
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