IMO Shortlist 1995 problem G3
Dodao/la:
arhiva2. travnja 2012. The incircle of triangle
![\triangle ABC](/media/m/1/f/3/1f3c3c0f3e134a169655f9511ba6ea82.png)
touches the sides
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
at
![D, E, F](/media/m/e/c/a/ecac01ab91092791814283e516ec0b5a.png)
respectively.
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
is a point inside triangle of
![\triangle ABC](/media/m/1/f/3/1f3c3c0f3e134a169655f9511ba6ea82.png)
such that the incircle of triangle
![\triangle XBC](/media/m/6/2/1/621a04dc3833f520ca09fd8890b56c52.png)
touches
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
, and touches
![CX](/media/m/3/e/9/3e92a8fed18c9f151c4930b9f011439c.png)
and
![XB](/media/m/e/8/d/e8d214d522136f014c7aa2406f65d930.png)
at
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
and
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
respectively.
Show that
![E, F, Z, Y](/media/m/9/3/4/93402086973a58a9acb9d0d8319a7b9f.png)
are concyclic.
%V0
The incircle of triangle $\triangle ABC$ touches the sides $BC$, $CA$, $AB$ at $D, E, F$ respectively. $X$ is a point inside triangle of $\triangle ABC$ such that the incircle of triangle $\triangle XBC$ touches $BC$ at $D$, and touches $CX$ and $XB$ at $Y$ and $Z$ respectively.
Show that $E, F, Z, Y$ are concyclic.
Izvor: Međunarodna matematička olimpijada, shortlist 1995