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Berkeley Math Circle Monthly Contest 4

Geometry Difficulty 4.4 AIME Prove it United States

Problem:
A triangle, two of whose sides are $3$ and $4$ , is inscribed in a circle. Find the minimal possible radius of the circle.

Solution

Solution:
Since the circle has a chord of length $4$ , its diameter is at least $4$ and so its radius is at least $2$ . To achieve equality, choose a right triangle with hypotenuse $4$ and one leg $3$ (the other leg will, by the Pythagorean theorem, have length $\sqrt{7}$ ). Then the midpoint of the hypotenuse is the center of a circle of radius $2$ passing through all three vertices.

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