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 2 of 3: Two symmetries
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AB⊥EM,M is the midpoint of PQ.AB\perp EM,\qquad M\text{ is the midpoint of }PQ.
Detailed analysis

The equal angle relations make AB the perpendicular bisector of EM; the common-chord midpoint transfers to PQ.