1982 USAMO Problems/Problem 5
, and are three interior points of a sphere such that and are perpendicular to the diameter of through , and so that two spheres can be constructed through , , and which are both tangent to . Prove that the sum of their radii is equal to the radius of .
Let the two tangent spheres be and , and let and be the origins and radii of respectively. Then stands normal to the plane through . Because both spheres go through , , and , the line also stands normal to , meaning and are both coplanar and parallel. Therefore the problem can be flattened to the plane through , , and .
Let be points on such that
Let be the three circles radical center, meaning and are tangent segments to and .
Because we have that and are diameters.
This means that and .
And because and we have that and .
We then conclude that
Let denote line bisecting line segment .
Since it follows that . Similarly we have that .
And so is a parallelogram because and bisect each other, meaning
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