KöMaL 2026 A927

Let $ABCDEF$ be a bicentric hexagon, that is a hexagon which is both cyclic and tangential. Assume that there exists a point $P$ such that line $AP$ is perpendicular to line $BF$, $CP$ is perpendicular to $BD$, and $EP$ is perpendicular to $DF$. Prove that there are two opposite sides of the hexagon whose sum is equal to the circumdiameter.