Definition:Circumscribe

Circle

A circle is circumscribed around a polygon when it passes through each of the vertices of that polygon:

Note that such a polygon can also be described as being inscribed in the circle.

Not all polygons can be circumscribed.

Here is an example of one that can not:

All triangles can be uniquely circumscribed.

Euclid's Terminology

As Euclid defined it:

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

(The Elements: Book IV: Definition $3$)

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.

(The Elements: Book IV: Definition $6$)

Rectilineal Figure

As Euclid defined it:

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.

(The Elements: Book IV: Definition $2$)