An acute triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
is given. Points
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
and
![A_2](/media/m/a/2/5/a25c6dade4a684fc874981a7d65625f5.png)
are taken on the side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
(with
![A_2](/media/m/a/2/5/a25c6dade4a684fc874981a7d65625f5.png)
between
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
),
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
and
![B_2](/media/m/1/8/1/181de00f42000a442a347ff370e521f1.png)
on the side
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
(with
![B_2](/media/m/1/8/1/181de00f42000a442a347ff370e521f1.png)
between
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
and
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
), and
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
and
![C_2](/media/m/a/b/8/ab898e857261e1c35339f3f3d8362ba0.png)
on the side
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
(with
![C_2](/media/m/a/b/8/ab898e857261e1c35339f3f3d8362ba0.png)
between
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
) so that
![\angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1.](/media/m/3/9/4/3942c6d6f76f01e7434b8831265502ac.png)
The lines
![AA_1,BB_1,](/media/m/3/9/6/3964d61f6245d7bdf6bdd3584567df8e.png)
and
![CC_1](/media/m/0/8/0/0801dadadab90cd497baf071c549e706.png)
bound a triangle, and the lines
![AA_2,BB_2,](/media/m/4/8/a/48a9538f2912e7b93eb70a68a79941ff.png)
and
![CC_2](/media/m/a/9/8/a98de0b56e0d06c1c5ea4a9aea49ae0b.png)
bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
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An acute triangle $ABC$ is given. Points $A_1$ and $A_2$ are taken on the side $BC$ (with $A_2$ between $A_1$ and $C$), $B_1$ and $B_2$ on the side $AC$ (with $B_2$ between $B_1$ and $A$), and $C_1$ and $C_2$ on the side $AB$ (with $C_2$ between $C_1$ and $B$) so that
$$\angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1.$$
The lines $AA_1,BB_1,$ and $CC_1$ bound a triangle, and the lines $AA_2,BB_2,$ and $CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.