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Problem 6

A convex quadrilateral ABCDABCD satisfies AB⋅CD=BC⋅DAAB\cdot CD=BC\cdot DA. Point XX lies inside ABCDABCD so that ∠XAB=∠XCD\angle XAB=\angle XCD and ∠XBC=∠XDA\angle XBC=\angle XDA. Prove that ∠BXA+∠DXC=180∘\angle BXA+\angle DXC=180^\circ.
Step 5 of 8: Use quasi-harmonic uniqueness
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D′A′B′C′∼ABCDD'A'B'C'\sim ABCD
Detailed analysis

The original quadrilateral and the relabelled inverse image are both convex quasi-harmonic quadrilaterals, and Step 4 gives the same four cyclic angles. By the uniqueness lemma, there is a similarity sending D′,A′,B′,C′D',A',B',C' respectively to A,B,C,DA,B,C,D.