Competition problem — Geometry — Maths Olympiad Prep

Geometry Difficulty 8.3 Shortlist Prove it India

Let $H$ be the orthocenter of triangle $ABC$ . Let $E, F$ be the feet of the $B, C$ -altitudes. Let $D, M, N$ be the midpoints of segments $AH, BD, CD$ respectively, and $T$ be the intersection of line $FM$ and $EN$ . Suppose $D, E, T$ , and $F$ are concyclic. Prove that $DT$ passes through the circumcentre of $ABC$ .

Solution

Let $O$ be the circumcenter of $(ABC)$ and $J$ be the midpoint of $DO$ . Now it is sufficient to prove that $TD$ and $TJ$ coincide. We first prove that $M, N, T, J$ are concyclic.
$\angle MJN = \angle BOC = 2\angle BAC = \angle EDF = 180^\circ - \angle FTE = \angle 180^\circ - \angle NJM$
This proves our claim!

Now note that $JM = \frac{OB}{2} = \frac{OC}{2} = JN$ , so $TJ$ is the angle bisector of $\angle NTM = \angle ETF$ . But, $D$ is the midpoint of arc $EF$ in the nine-point circle, so $TD$ is the angle bisector of $\angle ETF$ as well. Thus, $TD$ and $TJ$ must coincide! $\square$

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