The Euclid's Elements of Geometry

Geometry is the study of figures. Euclid's Elements provides themost fundamental way of learning geometry geometrically.

Book IV: Constructions for inscribed and circumscribed figures

Definitions

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

Propositions

1. To fit into a given circle a straight line equal to a given straight line which is not greater than the diameter of the circle.
2. To inscribe in a given circle a triangle equiangular with a given triangle.
3. To circumscribe about a given circle a triangle equiangular with a given triangle.
4. To inscribe a circle in a given triangle.
5. To circumscribe a circle about a given triangle.
   Corollary: When the center of the circle falls within the triangle, the triangle is acute-angled; when the center falls on a side, the triangle is right-angled; and when the center of the circle falls outside the triangle, the triangle is obtuse-angled.
6. To inscribe a square in a given circle.
7. To circumscribe a square about a given circle.
8. To inscribe a circle in a given square.
9. To circumscribe a circle about a given square.
10. To construct an isosceles triangle having each of the angles at the base double the remaining one.
11. To inscribe an equilateral and equiangular pentagon in a given circle.
12. To circumscribe an equilateral and equiangular pentagon about a given circle.
13. To inscribe a circle in a given equilateral and equiangular pentagon.
14. To circumscribe a circle about a given equilateral and equiangular pentagon.
15. To inscribe an equilateral and equiangular hexagon in a given circle.
    Corollary: The side of the hexagon equals 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 similar to those explained in the case of the pentagon wecan both inscribe a circle in a given hexagon and circumscribe one about it.
16. To inscribe an equilateral and equiangular fifteen-angled figure in a given circle.
    Corollary: 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 a fifteen-angled figure which is equilateral and equiangular. And further, by proofs similar to those in the case of the pentagon, we can bothinscribe a circle in the given fifteen-angled figure and circumscribe one aboutit.