1. Show that the largest circle that fits inside a triangle whose sides have lengths 3, 4, 5 has radius 1.
2. Show that the largest circle that fits inside a triangle whose sides have lengths 5, 12, 13 has radius 2.

Can you find a right-angled triangle such that the largest circle that fits inside it has radius 3? Of course, one such triangle has sides of lengths 9,12,15 (which is obtained by scaling the 3,4,5 triangle by a factor 3) but can you find another? What about an inradius of 4 or 5 or ...?