Triangle ABC has coordinates A(0, 0) B(23, 0), and vertex C lying in Quadrant I. The length of side BC = 11 and the sin(angle ABC = 0.768) when rounded to the nearest thousandth. Circle Q is drawn such that each side of triangle ABC is tangent to circle Q at exactly one point, and circle Q is inscribed within triangle ABC. If F is the x-coordinate of point Q, G is the y-coordinate of point Q, and H is the length of the diameter of circle Q, then a/b = (F + G^2 + H^2), where a and b are coprime positive integers. Find a + b.