AIME 2024 / AIME 2024 train cedfddc6-965b-5cbd-bddd-0a83e0fb41c3

Let ABC be a triangle inscribed in circle ω. Let the tangents to ω at B and C intersect at point D, and let (AD)¯ intersect ω at P. If AB=5, BC=9, and AC=10, AP can be written as the form (m)/(n), where m and n are relatively prime integers. Find m+n.

[asy] import olympiad;  unitsize(15);  pair A, B, C, D, E, F, P, O;  C = origin; A = (10,0); B = (7.8, 4.4899); draw(A--B--C--cycle); draw(A..B..C..cycle, red+dotted);  O = circumcenter(A, B, C);  E = rotate(90,B) * (O); F = rotate(90,C) * (O);  D = IP(B..E + (B-E)*4, C..F + (C-F)*-3);  draw(B--D--C--D--A);  P = IP(D..A, A..B..C);  dot(A); dot(B); dot(C); dot(D); dot(P); label("A", A, dir(335)); label("B", B, dir(65)); label("C", C, dir(200)); label("D", D, dir(135)); label("P", P, dir(235)); [/asy]

Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$.

[asy] import olympiad;  unitsize(15);  pair A, B, C, D, E, F, P, O;  C = origin; A = (10,0); B = (7.8, 4.4899); draw(A--B--C--cycle); draw(A..B..C..cycle, red+dotted);  O = circumcenter(A, B, C);  E = rotate(90,B) * (O); F = rotate(90,C) * (O);  D = IP(B..E + (B-E)*4, C..F + (C-F)*-3);  draw(B--D--C--D--A);  P = IP(D..A, A..B..C);  dot(A); dot(B); dot(C); dot(D); dot(P); label("$A$", A, dir(335)); label("$B$", B, dir(65)); label("$C$", C, dir(200)); label("$D$", D, dir(135)); label("$P$", P, dir(235)); [/asy]