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 5 of 5: Finish by directed-angle telescoping

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\angle O_1AO_2=\angle M_1AM_2

Detailed analysis

Writing the angle from $AO_1$ to $AO_2$ through the intermediate rays $AM_1,AM_2$ , the two equal correction angles from Step 4 cancel: $\angle(AO_1,AO_2)=\angle(AO_1,AM_1)+\angle(AM_1,AM_2)+\angle(AM_2,AO_2)=\angle(AM_1,AM_2)$ . Hence $\angle O_1AO_2=\angle M_1AM_2$ .