can you find the isosceles triangle of smallest area that circumscribes a circle of radius of one?

It is actually easier to do the problem in reverse.

Take an isosceles triangle. Find the largest circle which is inside of this. [There are several natural ways to do this, but the obvious one is the methods for the 'incircle of a triangle', using the angle bisectors to find the center.]

Now think about scaling this pair - so the circle has radius 1. The isosceles triangle you get will be the smallest possible one.

Having 'seen' what should happen (perhaps with something like Geometers Sketchpad) you can recognize the pattern of the triangle around the circle. With this image in mind (e.g. the angles at the center of the circle to the points of tangency of the the triangle sides) you can work out from a unit circle to the isoscles triangle.

All that 'smallest area' means here is - have the sides of the triangle tangent to the circle.

Walter Whiteley York University