2017 AIME II Problems/Problem 12
Circle
has radius
, and the point
is a point on the circle. Circle
has radius
and is internally tangent to
at point
. Point
lies on circle
so that
is located
counterclockwise from
on
. Circle
has radius
and is internally tangent to
at point
. In this way a sequence of circles
and a sequence of points on the circles
are constructed, where circle
has radius
and is internally tangent to circle
at point
, and point
lies on
counterclockwise from point
, as shown in the figure below. There is one point
inside all of these circles. When
, the distance from the center
to
is
, where
and
are relatively prime positive integers. Find
.
[asy]
draw(Circle((0,0),125));
draw(Circle((25,0),100));
draw(Circle((25,20),80));
draw(Circle((9,20),64));
dot((125,0));
label("
",(125,0),E);
dot((25,100));
label("
",(25,100),SE);
dot((-55,20));
label("
",(-55,20),E);
[/asy]