Let the excircle of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
lying opposite to
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
touch its side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at the point
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
. Define the points
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
and
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
analogously. Suppose that the circumcentre of the triangle
![A_1B_1C_1](/media/m/1/a/f/1af9d15fbb4b582c4f99670f42359e2d.png)
lies on the circumcircle of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Prove that the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
is right-angled.
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Let the excircle of the triangle $ABC$ lying opposite to $A$ touch its side $BC$ at the point $A_1$. Define the points $B_1$ and $C_1$ analogously. Suppose that the circumcentre of the triangle $A_1B_1C_1$ lies on the circumcircle of the triangle $ABC$. Prove that the triangle $ABC$ is right-angled.