MathLabs

Problem 1

Let ABCDABCD be a quadrilateral with all sides equal and ∠ABC=60∘\angle ABC=60^\circ. Let ℓ\ell be a line through DD that does not meet the quadrilateral except at DD. Let E,FE,F be the intersections of ℓ\ell with AB,BCAB,BC, respectively, and let M=CE∩AFM=CE\cap AF. Prove that CA2=CM⋅CECA^2=CM\cdot CE.
Step 4 of 5: Translate similarity into a tangent angle
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∠CAM=∠CAF=∠AEC\angle CAM=\angle CAF=\angle AEC
Detailed analysis

The point MM lies on AFAF, so ∠CAM=∠CAF\angle CAM=\angle CAF. The similarity △EAC∼△ACF\triangle EAC\sim\triangle ACF gives ∠CAF=∠AEC\angle CAF=\angle AEC. Hence ∠CAM=∠AEC\angle CAM=\angle AEC, an angle subtended by chord AMAM in the circumcircle of AMEAME.