Let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
be one of the two distinct points of intersection of two unequal coplanar circles
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
and
![C_2](/media/m/a/b/8/ab898e857261e1c35339f3f3d8362ba0.png)
with centers
![O_1](/media/m/7/2/b/72b270d556043f6f393afbf50620eb57.png)
and
![O_2](/media/m/f/2/d/f2de7ab4fb5625160a4d2f2ac2dd707d.png)
respectively. One of the common tangents to the circles touches
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
at
![P_1](/media/m/a/8/8/a886eaf7832af6b6b5f56f0ec9a97490.png)
and
![C_2](/media/m/a/b/8/ab898e857261e1c35339f3f3d8362ba0.png)
at
![P_2](/media/m/e/c/8/ec8662164615835e6c2307d72a487ec8.png)
, while the other touches
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
at
![Q_1](/media/m/0/3/1/0313e7ec2e52d7e7514e810cd41daf66.png)
and
![C_2](/media/m/a/b/8/ab898e857261e1c35339f3f3d8362ba0.png)
at
![Q_2](/media/m/7/5/f/75f4681412f8aab76f96fc5b7786365f.png)
. Let
![M_1](/media/m/9/7/2/9728b6a1cf3e905234b6989eae0cc038.png)
be the midpoint of
![P_1Q_1](/media/m/e/d/6/ed6d1cff823e8564e9ac22bb8940927d.png)
and
![M_2](/media/m/4/7/1/4718b55cbd1692b474c94c4104ddb007.png)
the midpoint of
![P_2Q_2](/media/m/4/6/e/46e109dcf6c27c050e398d6eea93eb70.png)
. Prove that
![\angle O_1AO_2=\angle M_1AM_2](/media/m/3/a/b/3ab1da48904f4f8cb5396344b8243f47.png)
.
%V0
Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.