2024 SSMO Team Round Problems/Problem 9

Problem

Let $ABCDEFGH$ be an equiangular octagon such that $AB=6, BC=8, CD=10, DE=12, EF=6, FG=8, GH=10,$ and $AH=12$. The radius of the largest circle that fits inside the octagon can be expressed as $a+b\sqrt{c},$ where \(a,b,\) and \(c\) are integers and \(c\) is squarefree. Find $a+b+c.$

Solution