Problem 2

Let

A

be one of the two distinct intersection points of unequal coplanar circles

C_1,C_2

with centers

O_1,O_2

. One common tangent touches them at

P_1,P_2

, and the other at

Q_1,Q_2

. Let

M_1,M_2

be the midpoints of

P_1Q_1,P_2Q_2

. Prove that

\angle O_1AO_2=\angle M_1AM_2

Step 1 of 5: Introduce the homothety center

S

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S=P_1P_2\cap Q_1Q_2,\qquad S\in O_1O_2

Detailed analysis

The two common tangents meet at $S$ . The center line of two nonconcentric circles passes through the external homothety center, so $S,O_1,O_2$ are collinear.