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Incircles
Age 16 to 18
Challenge Level 
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(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.
(3) 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?
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What about an inradius of 4 or 5 or ...?
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