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Problem 1041

AMC 12 late, AIME early

Geometry Difficulty 4.9 Prove it Romanian Mathematical Olympiad · Romania

A triangle $ABC$ has its orthocenter $H$ distinct from its vertices and from the circumcenter $O$ . Denote $M, N, P$ the circumcenters of the triangles $HBC, HCA$ , respectively $HAB$ . Prove that the lines $AM, BN, CP$ and $OH$ are concurrent.
Petru Braica

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

If $D$ is the midpoint of $[BC]$ , then $\overrightarrow{OD} = \frac{1}{2}(\overrightarrow{OB} + \overrightarrow{OC}) = \frac{1}{2}(\overrightarrow{OH} - \overrightarrow{OA}) = \frac{1}{2}\overrightarrow{AH}$ , hence $\overrightarrow{OM} = \overrightarrow{AH}$ .
It follows that $AHMO$ is a parallelogram, hence $AM$ passes through the midpoint of the segment $[OH]$ , the same being true for $BN$ and $CP$ .

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