MathLabs

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 1 of 8: Name the side-product condition
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In plain words

The hypothesis is a multiplicative symmetry between opposite sides.

AB⋅CD=BC⋅DAAB\cdot CD=BC\cdot DA
Detailed analysis

Call a convex quadrilateral satisfying AB⋅CD=BC⋅DAAB\cdot CD=BC\cdot DA quasi-harmonic. We first use the standard uniqueness lemma: a quasi-harmonic quadrilateral is determined up to similarity by its four (cyclically ordered) angles.