What is the simplest way to prove that the ratio of diameter and circumference of any circle is the same number?
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Draw two concentric circles, and draw a regular n-gon in one of them. Connect the centre of the circles to the vertices of that n-gon by semilines the circle and label the intersection of those with the other circle. You get a similar n-gon.
Similarity shows that the ratios of the perimeters of the two n-gons is the ratio of the diameters of the circles; and the circumference is defined as the limit of the perimeter of n-gons.