MathDB

9.6

Part of 2023 Sharygin Geometry Olympiad

Problems(1)

prove AB'C' and PQN are tangent

Source: Sharygin Finals 2023 9.6

8/2/2023
Let ABCABC be acute-angled triangle with circumcircle Γ\Gamma. Points HH and MM are the orthocenter and the midpoint of BCBC respectively. The line HMHM meets the circumcircle ω\omega of triangle BHCBHC at point N≠HN\not= H. Point PP lies on the arc BCBC of ω\omega not containing HH in such a way that ∠HMP=90∘\angle HMP = 90^\circ. The segment PMPM meets Γ\Gamma at point QQ. Points B′B' and C′C' are the reflections of AA about BB and CC respectively. Prove that the circumcircles of triangles AB′C′AB'C' and PQNPQN are tangent.
geometrySharygin Geometry OlympiadSharygin 2023tangent circlesorthocenter