A convex quadrilateral has perpendicular diagonals. The perpendicular bisectors of the sides and meet at a unique point inside . Prove that the quadrilateral is cyclic if and only if triangles and have equal areas.
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A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.
Source: Međunarodna matematička olimpijada, shortlist 1998
Školjka 
has perpendicular diagonals. The perpendicular bisectors of the sides 
and 
meet at a unique point 
inside 
and 
have equal areas.