<ADO = 90 ^o wanted, diameter , 2midpoints, perpendiculars related

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Source: Mathematics Regional Olympiad of Mexico Center Zone 2009 P1

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#1 h Nov 10, 2021, 5:54 PM

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Let $\Gamma$ be a circle with the center $O$ and let $A$ , $A ^ \prime$ be two diametrically opposite points in $\Gamma$ . Let $P$ be the midpoint of $OA ^ \prime$ and $\ell$ a line that passes through $P$ , different from the line $AA ^ \prime$ and different from the line perpendicular on $AA ^ \prime$ . Let $B$ and $C$ be the intersection points of $\ell$ with $\Gamma$ , let $H$ be the foot of the altitude from $A$ on $BC$ , let $M$ be the midpoint of $BC$ , and let $D$ be the intersection of the line $A ^ \prime M$ with $AH$ . Show that the angle $\angle ADO = 90 ^ \circ$ .

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#2 h Nov 10, 2021, 6:36 PM

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$AH ||OM$ since $M$ is midpoint of $BC$ $\implies$ $AD=2OM$ $\implies$ $D$ is orthocenter of $\triangle ABC$ $\implies$ $D-M-A'$ are collinear and $DM=MA'$ $\implies$ $DO||MP$ $\implies$ $\angle DOM=90$ $\implies$ $\angle ODH=90$ $\implies$ $\angle ADO=90$ .

This post has been edited 1 time. Last edited by Jalil_Huseynov, Nov 10, 2021, 6:37 PM

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