Mock AIME I 2012 Problems/Problem 7
Problem
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
.
Solution
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 .