Let
be one of the two distinct points of intersection of two unequal coplanar circles
and
with centers
and
respectively. One of the common tangents to the circles touches
at
and
at
, while the other touches
at
and
at
. Let
be the midpoint of
and
the midpoint of
. Prove that
.
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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 $\angle O_1AO_2=\angle M_1AM_2$.
Školjka