OB=OC wanted, 2 circumcircles, 1 midpoint, orthocenter

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OB=OC wanted, 2 circumcircles, 1 midpoint, orthocenter

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Source: Mathematics Regional Olympiad of Mexico West 2020 P6

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#1 h Sep 9, 2022, 8:25 PM

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Let $M$ be the midpoint of side $BC$ of a scalene triangle $ABC$ . The circle passing through $A$ , $B$ and $M$ intersects side $AC$ again at $D$ . The circle passing through $A$ , $C$ and $M$ cuts side $AB$ again at $E$ . Let $O$ be the circumcenter of triangle $ADE$ . Prove that $OB=OC$ .

This post has been edited 1 time. Last edited by parmenides51, Sep 9, 2022, 8:35 PM

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#2 h Oct 9, 2022, 7:56 AM

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This is just angle chasing. And until the very last step $M$ can be anywhere on $BC$ .

Let $T$ be the intersection of $BD$ and $CE$ .

Angle chase to show angle $ETD$ equals angle $BAC$ . Hence $T$ lies on circle $(ADE)$ ,
and also points $M,D,T,C$ lie on the same circle.

Let $P$ be the circumcentre of $(ADMB)$ and $Q$ be the circumcentre of $(MDTC)$ .
Angle chase to show that angle $POQ$ plus angle $PMQ$ sum to $180$ .
So $O,P,M,Q$ lie on the same circle, and hence $OM$ is perpendicular to $BC$

Now, finally using that $M$ is the midpoint of $BC$ , the conclusion follows

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