Kepler-Bouwkamp constant

Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex. The sum of the radii of the circles inscribed in these triangles is the same independent of the polygon vertex chosen. If a triangle is inscribed in a circle, another circle inside the triangle, a square inside the circle, another circle inside the square, and so on. Then the equation relating the inradius and circumradius of a regular polygon, r = R cos(π/n) gives the ratio of the radii of the final to initial circles as K' congruent r_(final circle)/r_(initial circle) = cos(π/3) cos(π/4) cos(π/5)....

cyclic polygon | infinite product | nested polygon | polygon circumscribing | whirl