Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle,
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
its incircle and
![\Omega_{a}, \Omega_{b}, \Omega_{c}](/media/m/1/7/a/17a057fee7b2360aeeba2a985af69134.png)
three circles orthogonal to
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
passing through
![(B,C),(A,C)](/media/m/3/b/3/3b344eaae059b9968ef864ab298d3e3c.png)
and
![(A,B)](/media/m/d/9/4/d94c637e9b52dceae5bac4741abe967c.png)
respectively. The circles
![\Omega_{a}](/media/m/6/0/0/60044068aedc043888943f7f6e2740bc.png)
and
![\Omega_{b}](/media/m/d/f/f/dffd95810ce83b3316d99ecbb2009e0f.png)
meet again in
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
; in the same way we obtain the points
![B'](/media/m/a/1/a/a1a88eb7f35fee4f41c66bfb0c902f51.png)
and
![A'](/media/m/9/2/6/9267b8bcabe1ad2df0d51dab3364714b.png)
. Prove that the radius of the circumcircle of
![A'B'C'](/media/m/5/3/d/53d1d147ad89bd52a7038ce57a0957ef.png)
is half the radius of
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
.
%V0
Let $ABC$ be a triangle, $\Omega$ its incircle and $\Omega_{a}, \Omega_{b}, \Omega_{c}$ three circles orthogonal to $\Omega$ passing through $(B,C),(A,C)$ and $(A,B)$ respectively. The circles $\Omega_{a}$ and $\Omega_{b}$ meet again in $C'$; in the same way we obtain the points $B'$ and $A'$. Prove that the radius of the circumcircle of $A'B'C'$ is half the radius of $\Omega$.