Points
![A_{1}](/media/m/9/7/4/9742b2655cd943b758073e1f1d090c23.png)
,
![B_{1}](/media/m/f/a/5/fa55cf39f6736c287bf64ee9471f00f1.png)
,
![C_{1}](/media/m/e/4/6/e46111370b6102ad343bcdc7190d9ff9.png)
are chosen on 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)
of a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
respectively. The circumcircles of triangles
![AB_{1}C_{1}](/media/m/3/2/b/32b5ec66877cd6e9615f0964958db4f8.png)
,
![BC_{1}A_{1}](/media/m/b/4/6/b4678dc433222877d2ace28881fc009a.png)
,
![CA_{1}B_{1}](/media/m/5/d/f/5df0c56117933a594ea8d9670acdc858.png)
intersect the circumcircle of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
again at points
![A_{2}](/media/m/a/2/5/a25771e5c2a6c9c6113eab3c46cf63d8.png)
,
![B_{2}](/media/m/5/f/0/5f0ace33ca787cd757fffa052940b6a9.png)
,
![C_{2}](/media/m/1/3/c/13ce248ee56b45a1b9e032736b8100a1.png)
respectively (
![A_{2}\neq A, B_{2}\neq B, C_{2}\neq C](/media/m/c/6/1/c61f14e8e0bcfc4df78ab18b9fb203e7.png)
). Points
![A_{3}](/media/m/3/8/e/38ec7244b139c312a2f4c1dcb9916272.png)
,
![B_{3}](/media/m/2/5/0/2502403995c0d8b205d6b2aec3f9b40a.png)
,
![C_{3}](/media/m/d/c/9/dc938352824cc502d8ecef86dc228222.png)
are symmetric to
![A_{1}](/media/m/9/7/4/9742b2655cd943b758073e1f1d090c23.png)
,
![B_{1}](/media/m/f/a/5/fa55cf39f6736c287bf64ee9471f00f1.png)
,
![C_{1}](/media/m/e/4/6/e46111370b6102ad343bcdc7190d9ff9.png)
with respect to the midpoints of 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)
respectively. Prove that the triangles
![A_{2}B_{2}C_{2}](/media/m/b/5/5/b550a1fdb37f6d9a57f090911f1b019f.png)
and
![A_{3}B_{3}C_{3}](/media/m/f/2/e/f2e41f7d9e031663c56b7849c3d60a2c.png)
are similar.
%V0
Points $A_{1}$, $B_{1}$, $C_{1}$ are chosen on the sides $BC$, $CA$, $AB$ of a triangle $ABC$ respectively. The circumcircles of triangles $AB_{1}C_{1}$, $BC_{1}A_{1}$, $CA_{1}B_{1}$ intersect the circumcircle of triangle $ABC$ again at points $A_{2}$, $B_{2}$, $C_{2}$ respectively ($A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $A_{3}$, $B_{3}$, $C_{3}$ are symmetric to $A_{1}$, $B_{1}$, $C_{1}$ with respect to the midpoints of the sides $BC$, $CA$, $AB$ respectively. Prove that the triangles $A_{2}B_{2}C_{2}$ and $A_{3}B_{3}C_{3}$ are similar.