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