2025 HMMT November General Round Problem 8

Let \Gamma_1 and \Gamma_2 be two circles that intersect at two points P and Q. Let \ell_1 and \ell_2 be the common external tangents of \Gamma_1 and \Gamma_2. Let \Gamma_1 touch \ell_1 and \ell_2 at U_1 and U_2, respectively, and let \Gamma_2 touch \ell_1 and \ell_2 at V_1 and V_2, respectively. Given that PQ = 10 and the distances from P to \ell_1 and \ell_2 are 3 and 12, respectively, compute the area of the quadrilateral U_1U_2V_2V_1.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2025 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.