2023 WSMO Team Round Problems/Problem 9

Problem

A circle has three chords of equal length, $4 + 2 \sqrt{3}$ which intersect forming a triangle with side lengths 2, 2, and $2 \sqrt{3}$ . If the radius of the circle is $r,$ then $r^2 = a-b\sqrt{c},$ for positive integers $a,b$ and squarefree $c.$ Find $a+b+c.$

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2023 WSMO Team Round Problems/Problem 9

Problem

Solution