USAMO 2011 Problem #5
by KingSmasher3, Apr 12, 2013, 11:39 PM
Let 
be a given point inside quadrilateral 
. Points 
and 
are located within such that
$\begin{align*}\angle Q_1BC&=\angle ABP,&\angle Q_1CB&=\angle DCP,& \angle Q_2AD&=\angle BAP,& \angle Q_2DA&=\angle CDP.\end{align*}$
Prove that
if and only if 
.
_______
If
then obviously we are done. If not, then let 
be the intersection of 
and 
Without loss of generality let be close to 
and 
than 
and 
Now note that from the given angles, is the isogonal conjugate of in 
and the isogonal conjugate of in 
This implies that both and lie on the reflection of 
over the angle bisector of 
In other words, 
are collinear. However, this is a contradiction if either or 
Therefore, if either orif 
so we are done.
Main Point(s): When given the angle configuration as in this problem, isogonal conjugates should come to mind.
be a given point inside quadrilateral 
. Points 
and 
are located within such that$\begin{align*}\angle Q_1BC&=\angle ABP,&\angle Q_1CB&=\angle DCP,& \angle Q_2AD&=\angle BAP,& \angle Q_2DA&=\angle CDP.\end{align*}$
Prove that
if and only if 
._______
If
then obviously we are done. If not, then let 
be the intersection of 
and 
Without loss of generality let be close to 
and 
than 
and 
Now note that from the given angles, is the isogonal conjugate of in 
and the isogonal conjugate of in 
This implies that both and lie on the reflection of 
over the angle bisector of 
In other words, 
are collinear. However, this is a contradiction if either or 
Therefore,
if either orif 
so we are done.Main Point(s): When given the angle configuration as in this problem, isogonal conjugates should come to mind.
This post has been edited 1 time. Last edited by KingSmasher3, Apr 12, 2013, 11:39 PM
March 2014
October 2013