Excenters orthocenters concurrence and more...

lambruscokid

Let $ABC$ be a triangle and let $I$ be the incenter. $Ia$ $Ib$ and $Ic$ are the excenters opposite to points $A$ $B$ and $C$ respectively. Let $La$ be the line joining the orthocenters of triangles $IBC$ and $IaBC$ . Define $Lb$ and $Lc$ in the same way.
Prove that $La$ $Lb$ and $Lc$ are concurrent.

Daniel

**Posted:** Fri Aug 08, 2008 10:55 am

campos

this is a very nice problem! consider points $A',B',C'$ such that $ABC$ are the midpoints of $B'C', A'C', A'B'$ , respectively... it's easy to prove that $L_a$ and $II_a$ correspond by reflection across the midpoint of $BC$ ... since $A$ and $A'$ also correspond by reflection across the midpoint of $BC$ it follows that $L_a$ is the bisector of $\angle B'A'C'$ ... and the angle bisectors are concurrent, so we're done Mr. Green

sorry to be not too detailed but i think it's easy to prove all the things i said, if you want more details i'll explain it later...

**Posted:** Fri Aug 08, 2008 11:55 am

lambruscokid

Thank you very much!
I still have to check one of the reflections, but seems straight-forward ...
your solution is very very nice Mr. Green

Daniel

**Posted:** Fri Aug 08, 2008 1:05 pm