Inscribing Regular Hexagon in Circle/Porism

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Porism to Inscribing Regular Hexagon in Circle

In the words of Euclid:

From this it is manifest that the side of the hexagon is equal to the radius of the circle.

And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle an equilateral and equiangular hexagon in conformity with what was explained in the case of the pentagon.

And further by means to those explained in the case of the pentagon we can both inscribe a circle in a given hexagon and circumscribe one about it.

(The Elements: Book $\text{IV}$: Proposition $15$ : Porism)


In the same way as for the regular pentagon, we can draw tangents to the circle at the vertices of the regular hexagon.

This will draw a regular hexagon which has been circumscribed about the circle.

Further, in a similar way to methods used for the regular pentagon, a circle can be inscribed in a regular hexagon and circumscribed about a regular hexagon.