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Problem 1422

AIME late

Geometry Difficulty 5.3 Prove it Hong Kong competition problems · Hong Kong · 2016

Let $O$ be the circumcentre of a triangle $ABC$ , and let $\ell$ be the line going through the midpoint of the side $BC$ and is perpendicular to the bisector of $\angle BAC$ . Determine the value of $\angle BAC$ if the line $\ell$ goes through the midpoint of the line segment $AO$ .

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

We have $\angle BAC = 120^\circ$ .
Let $M$ be the midpoint of $BC$ . Suppose $l$ meets the altitude from $A$ , the angle bisector of $\angle BAC$ , and the line $AO$ at $E$ , $P$ , $N$ respectively.
Note that $AE // MO$ . In order that $EM$ bisects $AO$ , the quadrilateral $AEOM$ must be convex. Therefore, $\angle BAC$ must be obtuse.
Now, since $AE // MO$ and $AN = ON$ , we have $\triangle AEN \cong \triangle OMN$ . It is well-known that $AE$ and $AO$ are isogonal lines with respect to $\angle BAC$ . Therefore, $AP$ bisects $\angle EAN$ . As we also have $AP \perp EN$ , this shows $\triangle AEN$ is isosceles, with $AE = AN$ . Therefore,
$OM = AE = AN = \frac{1}{2} OA = \frac{1}{2} OB.$
As $\angle BMO = 90^\circ$ , this yields $\angle OBM = 30^\circ$ . Thus, $\angle BAC = 120^\circ$ .

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