Question on Triangles In a right triangle, the length of hypotenuse is $c$. The centers of three circles of radius $c/5$ are found at its vertices. Find the radius of the fourth circle which touches the three given circles and doesn't enclose them.

The midpoint of the hypotenuse is at a distance of $c/2$ from either of the endpoints of the hypotenuse. It's not hard to show that it's also at that same distance from the vertex of the right angle. Therefore a circle centered there that is just big enough to touch the two circles of radius $c/5$ centered at the endpoints of the hypotenuse will also be just big enough to touch that third circle of radius $c/5$ centered at the vertex of the right angle. So do the arithmetic.