1994 OIM Problems/Problem 2

Problem

Let there be a quadrilateral inscribed in a circle, whose vertices are denoted consecutively by $A$, $B$, $C$ and $D$. It is assumed that there exists a semicircle with center in $AB$, tangent to the other three sides of the quadrilateral.

i. Prove that $AB=AD+BC$

ii. Calculate, based on $x=AB$ and $y=CD$, the maximum area that a quadrilateral that satisfies the conditions of the statement can reach.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

https://www.oma.org.ar/enunciados/ibe9.htm