HMMT 2025 G6

Trapezoid $A B C D$ , with $A B\parallel C D$ , has side lengths $A B = 11$ , $B C = 8$ , $C D = 19$ , and $D A = 4$ . Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\triangle A B C$ , $\triangle B C D$ , $\triangle C D A$ , and $\triangle D A B$ .