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$ABCD$ is a convex quadrangle, and $$PA\cdot\sin A+PC\cdot\sin C=PB\cdot\sin B+PD\cdot\sin D.$$ Here $P = AC \cap BD$, $\sin A$ means $\sin \angle DAB$, $\sin B$ means $\sin \angle ABC$, $\sin C$ means $\sin \angle BCD$, $\sin D$ means $\sin \angle CDA$.

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Are $A,B,C,D$ concyclic?

Conversely, when $A,B,C,D$ are concyclic, it is easy to know $PA\cdot\sin A+PC\cdot\sin C=PB\cdot\sin B+PD\cdot\sin D$. But this, is it true?

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