The Most and the Least of Mickey

Geometry Level 5

Two lines intersect each other forming an angle \(a\). A circle of radius \(r\) is tangent to both of these lines. Two equal size circles with radius \(R\) are:

(1): tangent to this circle,
(2): tangent to each other, and
(3): each tangent to one of the lines,

As the size of the angle \(a\) changes, so does the value of the ratio \(\dfrac Rr\). If \(m\) and \(M\) are the infimum and the supremum of the ratio \(\dfrac Rr\) respectively, report \(m+M\).


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