A convex polygon with n sides is drawn in a circle. Then divided into triangles. Finally circles are inscribed in the triangles. Obviously, triangulation is not unique; a 4-gon has two, a 5-gon five, a 6-gon 14 and so on, as continue the Catalan numbers. But the sum of the radii of the circles is constant, independent of how the n-gon is divided.

A convex polygon with n sides is drawn in a circle. Then divided into triangles. Finally circles are inscribed in the triangles. Obviously, triangulation is not unique; a 4-gon has two, a 5-gon five, a 6-gon 14 and so on, as continue the Catalan numbers. But the sum of the radii of the circles is constant, independent of how the n-gon is divided.

Notes

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    The sum of the radii is independent of the method of polygonal decomposition.
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