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F is orthocentre of ABC iff HD||CF and H is on (ABC)

Source: Chinese MO 1999

March 19, 2012
geometrycircumcirclegeometry proposed

Problem Statement

Let ABCABC be an acute triangle with ∠C>∠B\angle C>\angle B. Let DD be a point on BCBC such that ∠ADB\angle ADB is obtuse, and let HH be the orthocentre of triangle ABDABD. Suppose that FF is a point inside triangle ABCABC that is on the circumcircle of triangle ABDABD. Prove that FF is the orthocenter of triangle ABCABC if and only if HD∣∣CFHD||CF and HH is on the circumcircle of triangle ABCABC.
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