MEMO 2014 ekipno problem 5
Dodao/la:
arhiva24. rujna 2014. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle with
![AB < AC](/media/m/e/6/3/e63999705031b9fe2418f3e7a4977479.png)
. Its incircle with centre
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
touches the sides
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
, and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
in the points
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
, and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
respectively. The angle bisector
![AI](/media/m/d/6/c/d6ca198d9456e8d2c2e0f932598e4a02.png)
intersects the lines
![DE](/media/m/a/c/d/acdf3f4d3c794d9a897484e9d216f5ec.png)
and
![DF](/media/m/3/d/d/3dd8b7899102ac0f0d215a5d87897f88.png)
in the points
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
respectively. Let
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
be the foot of the altitude through
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
with respect to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
.
Prove that
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
is the incentre of the triangle
![XYZ](/media/m/1/3/d/13dab5022dd1d33f3d299852f2f54cfb.png)
.
%V0
Let $ABC$ be a triangle with $AB < AC$. Its incircle with centre $I$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$, and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$.
Prove that $D$ is the incentre of the triangle $XYZ$.
Izvor: Srednjoeuropska matematička olimpijada 2014, ekipno natjecanje, problem 5