Proof That For All Circles The Circumference To Radius Ratio Is The Same

Here is a proof that, for all circles, the circumference to radius ratio is the same:

• Circles can be approximated by regular polygons composed of congruent isosceles triangles:
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• For similar polygons all pairs of triangles, for all pairs of circles, are similar.
• All altitude and side length ratios for all pairs of triangles, for all pairs of circles, are equal.
• All perimeter to inradius ratios for all similar polygons, for all pairs of circles, are equal.
• As the number of polygon sides increases, perimeter to inradius ratios approach circumference to radius ratios.
• Therefore, for all circles, the circumference to radius ratio is the same.
• QED