IMO Shortlist 2012 problem G4
Dodao/la:
arhiva3. studenoga 2013. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle with
![AB \neq AC](/media/m/3/f/8/3f81a154261c64790a8a9c47a02a9b3f.png)
and circumcenter
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
. The bisector of
![\angle BAC](/media/m/b/2/1/b21a9e466104c5d33646432221e142be.png)
intersects
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
. Let
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
be the reflection of
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
with respect to the midpoint of
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. The lines through
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
and
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
perpendicular to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
intersect the lines
![AO](/media/m/d/9/3/d93e4e1fde6437bd5210d0a50abb3ca8.png)
and
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
at
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
respectively. Prove that the quadrilateral
![BXCY](/media/m/f/8/f/f8fcaf887398deb3bdba89042c41d5a5.png)
is cyclic.
%V0
Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.
Izvor: Međunarodna matematička olimpijada, shortlist 2012