Nice geo
- Raiyan Jamil
- Posts:138
- Joined:Fri Mar 29, 2013 3:49 pm
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.
A smile is the best way to get through a tough situation, even if it's a fake smile.
Re: Nice geo
The first principle is that you must not fool yourself and you are the easiest person to fool.
- ahmedittihad
- Posts:181
- Joined:Mon Mar 28, 2016 6:21 pm
Re: Nice geo
$CH$ 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.
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.
Frankly, my dear, I don't give a damn.