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 4 of 8: Read the angle data after inversion
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∠A′=∠A,∠B′=∠B,∠C′=∠C,∠D′=∠D\angle A' =\angle A,\quad\angle B'=\angle B,\quad\angle C'=\angle C,\quad\angle D'=\angle D
Detailed analysis

Inversion preserves angles between lines (with the orientation reversed). Since XX lies inside the convex quadrilateral, the relabelled image D′A′B′C′D'A'B'C' has the same cyclic angle list as ABCDABCD: the angle at D′D' equals the original angle at AA, the angle at A′A' equals that at BB, and so on. We use this relabelling only to align the cyclic order.