Mock AIME I 2012 Problems/Problem 7
Let two circles with radii in the plane be centered at points , respectively. Consider a point in the plane such that . Denote the intersections of the line with as , and the intersections of the line with as . Let and intersect at points such that . If equals the area of , can be written in the form where are distinct squarefree integers. Find .
Let . Clearly we have that and . Let denote the power of with respect to the circles , respectively. Then . Additionally, , as well, so it follows that lies on the radical axis of the two circles. However, also lie on this radical axis, so are collinear.
Moreover, since the radical axis is perpendicular to , we have that is an altitude of the triangle . Notice that by considering the right triangles and using the Pythagorean Theorem we obtain that . It now suffices to find .
Since lies on the radical axis, we have that , so . So then the area can be written as .
This can be rewritten into the desired answer form, which is .