Euclid's Elements, Book IV, Definitions

Definitions

Def. 1. A rectilinear figure is said to be inscribed in a rectilinear figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.

Def. 2. Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.

Def. 3. A rectilinear figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.

Def. 4. A rectilinear figure is said to be circumscribed about a circle when each side of the circumscribed figure touches the circumference of the circle.

Def. 5. 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.

Def. 6. A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed.

Def. 7. A straight line is said to be fitted into a circle when its ends are on the circumference of the circle.

Guide

The first figure shows a smaller quadrilateral inscribed in a larger quadrilateral, and the larger one is circumscribed about the smaller one. The second figure shows a quadrilateral inscribed in a circle, and the circle is circumscribed about the quadrilateral. The third figure shows a circle inscribed in a quadrilateral, and the quadrilateral is circumscribed about the circle. Note also that in the second figure, each side of the quadrilateral is fitted into the circle.

Book IV Introduction - Proposition IV.1.