Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 7.6 National Olympiad, round 2 Prove it Hong Kong

Let $AD$ , $BE$ and $CF$ be respectively the altitudes of $\triangle ABC$ . Let $P$ , $Q$ and $R$ be points on the lines $BC$ , $CA$ and $AB$ respectively such that $AP$ is perpendicular to $EF$ , $BQ$ is perpendicular to $FD$ and $CR$ is perpendicular to $DE$ . Show that $AP$ , $BQ$ and $CR$ are concurrent.

Solution

By a corollary of Carnot's theorem, the perpendicular lines drawn from $A$ , $B$ , $C$ to $EF$ , $FD$ , $DE$ are concurrent if and only if the perpendicular lines drawn from $D$ , $E$ , $F$ to $BC$ , $CA$ , $AB$ are concurrent. The latter is clearly true since the perpendicular lines are just $AD$ , $BE$ , $CF$ , which are concurrent at the orthocentre of $\triangle ABC$ . This proves $AP$ , $BQ$ , $CR$ are concurrent.

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