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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 2 of 8: Prove the uniqueness lemma
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fixed angles+AB⋅CD=BC⋅DA⟹unique up to similarity\text{fixed angles}+AB\cdot CD=BC\cdot DA\Longrightarrow\text{unique up to similarity}
Detailed analysis

For completeness, normalize AB=1AB=1 and place A,BA,B on a fixed line. The prescribed angles determine the directions of the other two sides; writing their lengths as DA=tDA=t and BC=sBC=s, the sine rule in the two triangles cut by a diagonal gives one equation for each of the remaining angles. Eliminating the diagonal gives a strictly monotone equation in t/st/s. The side condition fixes CD/AB=(BC/DA)=s/tCD/AB=(BC/DA)=s/t, so it selects exactly one positive ratio and then one scale. Thus two convex quasi-harmonic quadrilaterals with the same four angles are similar. This is the elementary uniqueness lemma used in the official inversion solution.