The Most and the Least of Mickey
Two lines intersect each other forming an angle . A circle of radius is tangent to both of these lines. Two equal size circles with radius are:
(1)
: tangent to this circle,
(2)
: tangent to each other at a point more distant from the intersection of the lines than the center of the original circle, and
(3)
: each tangent to one of the lines,
As the size of the angle changes, so does the value of the ratio . If and are the infimum and the supremum of the ratio respectively, report .
The answer is 4.5.
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1 solution
The ratio R / r is an increasing function of angle a which varies from near zero to almost 180 degrees. As angle a approaches zero, the two lines are close to parallel and the circles approach situation pictured below:
The limit of the ratio R / r for parallel lines, that is m , is then 2 1
At the other extreme as angle a approaches 180 degrees, situation is pictured below:
Sides of the right triangle ABC are R , R + r , and R − r . The ratio R / r can be calculated from the Pythagorean Theorem and comes to M = 4 . So M + m = 4 . 5 .