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 8 of 8: Conclusion
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∠BXA+∠DXC=180∘\boxed{\angle BXA+\angle DXC=180^\circ}
Detailed analysis

The inversion and similarity force the isogonal-conjugacy relation, and Step 7 translates exactly that relation into ∠BXA+∠DXC=180∘\angle BXA+\angle DXC=180^\circ. This proves the claim.