In the convex quadrilateral $ABCD$, the bisectors of angles $A$ and $C$ intersect in $I$. Prove that $ABCD$ is circumscriptible if and only if $$S[AIB] + S[CID] =S[AID]+S[BIC]$$( $S[XYZ]$ denotes the area of the triangle $XYZ$)
Source: 1999 Romania NMO IX p3
Tags: geometry, tangential
In the convex quadrilateral $ABCD$, the bisectors of angles $A$ and $C$ intersect in $I$. Prove that $ABCD$ is circumscriptible if and only if $$S[AIB] + S[CID] =S[AID]+S[BIC]$$( $S[XYZ]$ denotes the area of the triangle $XYZ$)