Definition:Inscribe/Circle in Polygon

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A circle is inscribed in a polygon when it is tangent to each of the sides of that polygon:


In the words of Euclid:

Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed.

(The Elements: Book $\text{IV}$: Definition $5$)

Also known as

The more archaic language inscribed within is often seen.

Also see

Note that such a polygon can also be described as being circumscribed around the circle.

In the words of Euclid:

A rectilineal figure is said to be circumscribed about a circle, when each side of the circumscribed figure touches the circumference of the circle.

(The Elements: Book $\text{IV}$: Definition $4$)