Let be the centers and
the radii of the circles
. Let
be the points of tangency from the common external tangent of
, respectively, and let the extension of
intersect the extension of
at a point
. Let the endpoints of the chord/tangent be
, and the foot of the perpendicular from
to
be
. From the similar right triangles
,
It follows that , and that
.
By the Pythagorean Theorem on , we find that
and the answer is m+n+p=405