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PAMO 2022 Problem 1 - Line Tangent to Circle Through Orthocenter

Source: 2022 Pan-African Mathematics Olympiad Problem 1

June 25, 2022
geometrycircumcircle

Problem Statement

Let ABCABC be a triangle with ∠ABC≠90∘\angle ABC \neq 90^\circ, and ABAB its shortest side. Let HH be the orthocenter of ABCABC. Let Γ\Gamma be the circle with center BB and radius BABA. Let DD be the second point where the line CACA meets Γ\Gamma. Let EE be the second point where Γ\Gamma meets the circumcircle of the triangle BCDBCD. Let FF be the intersection point of the lines DEDE and BHBH.
Prove that the line BDBD is tangent to the circumcircle of the triangle DFHDFH.
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