Here is a decagon - a 10-sided regular polygon with all its angles equal and all its sides the same length - which has been divided into 10 triangles. Because of its symmetry, all the triangles have two sides that are the same length and so the two other angles in each triangle are also equal. In each triangle, what is the size of the angle at the centre of the decagon? We now know enough to identify the triangle since we know one angle and that the two sides surrounding it are equal. Which triangle on this page is it?

From what we have already found out about this triangle earlier, we can now say that

The radius of a circle through the points of a decagon is Phi times as long as the side of the decagon.

This follows directly from Euclid's Elements Book 13, Proposition 9.

1-61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More.. EJ

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