Problem
2025 AIME II Problem 6
Circle \omega_1 with radius 6 centered at point A is internally tangent at point B to circle \omega_2 with radius 15. Points C and D lie on \omega_2 such that \overline{BC} is a diameter of \omega_2 and {\overline{BC} \perp \overline{AD}}. The rectangle EFGH is inscribed in \omega_1 such that \overline{EF} \perp \overline{BC}, C is closer to \overline{GH} than to \overline{EF}, and D is closer to \overline{FG} than to \overline{EH}, as shown. Triangles \triangle {DGF} and \triangle {CHG} have equal areas. The area of rectangle EFGH is \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.
![[asy] size(5cm); defaultpen(fontsize(10pt)); pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5)); filldraw(G--H--C--cycle, lightgray); filldraw(D--G--F--cycle, lightgray); draw(B--C); draw(A--D); draw(E--F--G--H--cycle); draw(circle((0,0), 15)); draw(circle(A, 6)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); label("$A$", A, (.8, -.8)); label("$B$", B, (.8, 0)); label("$C$", C, (-.8, 0)); label("$D$", D, (.4, .8)); label("$E$", E, (.8, -.8)); label("$F$", F, (.8, .8)); label("$G$", G, (-.8, .8)); label("$H$", H, (-.8, -.8)); label("$\omega_1$", (9, -5)); label("$\omega_2$", (-1, -13.5)); [/asy]](/static/aops/4d17d40f7140b303cc92501c81e1fef807d9a2c8.png)
Leading zeroes must be inputted, so if your answer is 34, then input 034
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