Question 11
Let $ABCD$ be a cyclic quadrilateral with $AB = 4,$ $BC = 5,$ $CD = 6,$ and $DA = 7.$ Let $A_1$ and $C_1$ be the feet of the perpendiculars from $A$ and $C,$ respectively, to line $BD,$ and let $B_1$ and $D_1$ be the feet of the perpendiculars from $B$ and $D,$ respectively, to line $AC.$ The perimeter of $A_1B_1C_1D_1$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$