Given a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
satisfying
![AC+BC=3\cdot AB](/media/m/0/e/0/0e0f2d9b7f97f38b2d596494bc0a1fe4.png)
. The incircle of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
has center
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
and touches the sides
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
and
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
at the points
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
and
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
, respectively. Let
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
and
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
be the reflections of the points
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
and
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
with respect to
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
. Prove that the points
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
,
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
lie on one circle.
%V0
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.