MathLabs

Problem 1

Two circles Γ1\Gamma_1 and Γ2\Gamma_2 intersect at two points M M and N N . Let AB AB be the common tangent to these circles at A A and B B , respectively, so that M M lies closer to AB AB than N N . Let the line through M M parallel to AB AB meet Γ1\Gamma_1 again at C C and Γ2\Gamma_2 again at D D . Lines AC AC and BD BD meet at E E ; lines AN AN and CD CD meet at P P ; lines BN BN and CD CD meet at Q Q . Prove that EP=EQ EP=EQ .
Step 1 of 3: Equal tangent angles
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∠EAB=∠ACM=∠BAM,∠EBA=∠BDM=∠ABM.\angle EAB=\angle ACM=\angle BAM,\quad \angle EBA=\angle BDM=\angle ABM.
Detailed analysis

Apply the tangent-chord theorem and the parallelism of CD and AB.