### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

 If a ball is rolled into the corner of a room how far is its centre from the corner? A sphere of diameter $10$cm is packed into a cubic box whose internal side lengths are $12$cm. To prevent the sphere from rolling around 8 identical smaller spheres are placed in the corners of the box. What is the radius of one of the smaller spheres?