2000 AIME II Problems/Problem 12
The points , and lie on the surface of a sphere with center and radius . It is given that , , , and that the distance from to is , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .
It follows that , so is the circumcenter of .
From , we know that the circumradius of is:
Thus by the Pythagorean Theorem again,
So the final answer is .
Solution 2 (Vectors)
We know the radii to ,, and form a triangular pyramid . We know the lengths of the edges . First we can break up into its two component right triangles and . Let the axis be perpendicular to the base and axis run along , and occupy the other dimension, with the origin as . We look at vectors and . Since is isoceles we know the vertex is equidistant from and , hence it is units along the axis. Hence for vector , in form it is where is the height (answer) and is the component of the vertex along the axis. Now on vector , since is along , and it is along axis, it is . We know both vector magnitudes are . Solving for yields , so Answer = .
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this is highly trivial for an AIME #12