A BEAUTIFUL GEO

For discussing Olympiad level Geometry Problems
tanmoy
Posts:312
Joined:Fri Oct 18, 2013 11:56 pm
Location:Rangpur,Bangladesh
A BEAUTIFUL GEO

Unread post by tanmoy » Thu Jan 14, 2016 7:39 pm

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.
[$\text{Source}$:USA TST 2011,P7]
"Questions we can't answer are far better than answers we can't question"

User avatar
asif e elahi
Posts:185
Joined:Mon Aug 05, 2013 12:36 pm
Location:Sylhet,Bangladesh

Re: A BEAUTIFUL GEO

Unread post by asif e elahi » Mon Jan 25, 2016 10:32 pm

Take point $P'$ on ray $OQ$ so that $QB.QD=QO.QP=QE.QC$ where $Q=BD\cap CE$. Then $BODP'$ and $COEP'$ are cyclic.After some easy angle chasing, it can be shown that $\angle BP'C= \angle BAC=\angle EP'D$ which makes $ACBP'$ and $ADEP'$ cyclic. So $P'\equiv P$. The rest is easy.

Post Reply