A ball of radius 1 is in the corner of a room, tangent to two walls and the floor. A smaller ball is also in the corner, tangent to both walls, the floor, and the larger ball. The walls are perpendicular to each other, and they are perpendicular to the floor. Find the radius of the smaller ball.

Let the radius of the smaller sphere be Call the corner point the center of the big sphere and the center of the small sphere Note that is a space diagonal of a cube with side length (if this isn't clear drop perpendiculars from to the two walls and the floor), so Similarly, is a space diagonal of a cube with side length so it has length Note that passes through and thus also through the point of tangency of the two spheres; call this point We have as it is a radius of the smaller sphere and as it is a radius of the larger sphere. Therefore, we have the equation


Solving for gives