2013 Mock AIME I Problems/Problem 1
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Problem 1
Two circles and
, each of unit radius, have centers
and
such that
. Let
be the midpoint of
and let
be a circle externally tangent to both
and
.
and
have a common tangent that passes through
. If this tangent is also a common tangent to
and
, find the radius of circle
.
Solution
Let be the center of circle
and
be the point of tangency between
and
. Note that triangles
and
are similar, so
and
. Thus the radius of
is
.