# Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle

### (Corollary to Proposition 15 from Book 4 of Euclid's “Elements”)

So, from this, (it is) manifest that a side of the hexagon is equal to the radius of the circle. And similarly to a pentagon, if we draw tangents to the circle through the (sixfold) divisions of the (circumference of the) circle, an equilateral and equiangular hexagon can be circumscribed about the circle, analogously to the aforementioned pentagon. And, further, by (means) similar to the aforementioned pentagon, we can inscribe and circumscribe a circle in (and about) a given hexagon. (Which is) the very thing it was required to do.

### Modern Formulation

The side of a regular hexagon which is inscribed in a circle is equal to the radius of that circle. Moreover, by drawing a tangent to the circle through each of the edges of the inscribed regular hexagon, we get another regular hexagon, which is circumscribed about circle.

Proofs: 1

Sections: 1

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