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2 points concyclic with (BHC)

Source: 5th XMO (China) p1 https://artofproblemsolving.com/community/c3192772_geometry_regional_chin

April 26, 2024
geometryConcyclic

Problem Statement

Let △ABC\vartriangle ABC be an acute triangle with altitudes ADAD, BEBE, CFCF and orthocenter HH. Circle ⊙V\odot V is the circumcircle of △DEF\vartriangle DE F. Let segments FDFD, BHBH intersect at point PP. Let segments EDED, HCHC intersect at point QQ. Let KK be a point on ACAC such that VK⊥CFVK \perp CF. a) Prove that △PQH∼△AKV\vartriangle PQH \sim \vartriangle AKV. b) Let line PQPQ intersect ⊙V\odot V at points I,GI,G. Prove that points B,I,H,G,CB,I,H,G,C are concyclic with center the symmetric point XX of circumcenter OO of △ABC\vartriangle ABC wrt BCBC.
[hide=PS.] There is a chance that those in the hide were not wanted in the problem, as I tried to understand the wording from a solutions' video. I couldn't find the original wording pdf or picture.
https://cdn.artofproblemsolving.com/attachments/c/3/0b934c5756461ff854d38f51ef4f76d55cbd95.png
[url=https://www.geogebra.org/m/cjduebke]geogebra file
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