The adjoining figure shows two intersecting chords in a circle, with B on minor arc AD. Suppose that the radius of the circle is 5, that BC=6, and that AD is bisected by BC. Suppose further that AD is the only chord starting at A which is bisected by BC. It follows that the sine of the minor arc AB is a rational number. If this fraction is expressed as a fraction m/n in lowest terms, what is the product mn?
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1));
draw(Circle(origin, 1)^^A--D^^B--C);
real r=0.05;
pair M1=midpoint(M--D), M2=midpoint(M--A);
draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D)));
draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D)));
pair point=origin;
label("A", A, dir(point--A));
label("B", B, dir(point--B));
label("C", C, dir(point--C));
label("D", D, dir(point--D));[/asy] trigonometrygeometrygeometric transformationhomothetyratiorectangleanalytic geometry