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Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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Sangaku

The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.

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Escriptions

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

Incircles

Stage: 5 Challenge Level: Challenge Level:1

circle in triangle
(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?

What about an inradius of 4 or 5 or ...?