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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 6 of 8: Interpret the similarity as isogonal conjugacy
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the image of X is isogonal to X in ABCD\text{the image of }X\text{ is isogonal to }X\text{ in }ABCD
Detailed analysis

A standard inversion-similarity lemma says the following. If an interior point is inverted and the relabelled inverse quadrilateral is mapped back to the original one by a similarity, then the corresponding image of the inversion center has, at every vertex, the ray isogonal to the ray from that vertex to the original center. Applying it here, the point corresponding to the inversion center XX is an isogonal conjugate of XX in ABCDABCD.