Determining the radius of a circle tangent to two identical circles and a straight line tangent to all three. Here is a badly drawn MS Paint representation of the problem:

We have two circles, shown in red, each with radius $R$ and tangent to one another. A straight line, shown in purple, is drawn tangent to both red circles. We then draw a blue circle tangent to both red circles and the purple line. What is the radius of the blue circle, in terms of $R$?

Call the radius $r$. Then by Pythagorean Theorem $$R^2+(R-r)^2=(R+r)^2$$ so $$R^2=4rR\implies r={R\over 4}$$
