Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a convex quadrilateral. The diagonals
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
intersect at
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
. Show that
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
is cyclic if and only if
![AK \sin A + CK \sin C = BK \sin B + DK \sin D](/media/m/b/8/e/b8ea47088e3b46bb5a86fff987361462.png)
.
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Let $ABCD$ be a convex quadrilateral. The diagonals $AC$ and $BD$ intersect at $K$. Show that $ABCD$ is cyclic if and only if $AK \sin A + CK \sin C = BK \sin B + DK \sin D$.