# If $PA\sin A+PC\sin C=PB\sin B+PD\sin D$, are $A,B,C,D$ concyclic?

$$ABCD$$ 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 $$\sin \angle CDA$$.

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?