Nice geo

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Raiyan Jamil
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Nice geo

Unread post by Raiyan Jamil » Thu Dec 22, 2016 1:20 am

Let $P,Q$ be the feet of perpendiculars from the orthocentre $H$ on the internal and external angle bisectors of $A$ respectively of triangle $ABC$.Let $M$ be the midpoint of $BC$. Prove that $P,Q$ and $M$ are collinear.
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joydip
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Re: Nice geo

Unread post by joydip » Thu Dec 22, 2016 10:34 pm

Let $(O)$ be the circle with diameter $AH$.$D$ and $F$ be the projection of $C$ and $B$ on $AB$ and $AC$ respectively.
$A,F,P,D,Q \in (O).\angle QAP=90^\circ \Rightarrow Q,O,P$ are collinear. $\angle DAP =\angle PAF \Rightarrow OP$ bisect $DF$ .By Newton's theorem on $ADHF$, $OM$ bisect $DF \Rightarrow P,Q,M$ are collinear.
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ahmedittihad
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Re: Nice geo

Unread post by ahmedittihad » Sun Jan 01, 2017 2:23 pm

$CH$ meets $ AB$ at $E$, $BH$ meets $AC$ at $F$.
Let $X$ be the midpoint of $AH$. It suffices to prove that $X,P,M$ are colinear. Now, $X$ is the center of the circle with diameter $AH$. $P$ is the midpoint of arc $EF$. It's also well known that $ME , MF$ are tangent to the circle $AEF$. The result follows.
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