Euclid's Elements Book IV : Constructions for inscribed and circumscribed figures.

### Definitions

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

### Propositions

- Proposition 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.
- Proposition 2
- To inscribe in a given circle a triangle equiangular with a given triangle.
- Proposition 3
- To circumscribe about a given circle a triangle equiangular with a given triangle.
- Proposition 4
- To inscribe a circle in a given triangle.
- Proposition 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.

- Proposition 6
- To inscribe a square in a given circle.
- Proposition 7
- To circumscribe a square about a given circle.
- Proposition 8
- To inscribe a circle in a given square.
- Proposition 9
- To circumscribe a circle about a given square.
- Proposition 10
- To construct an isosceles triangle having each of the angles at the base double the remaining one.
- Proposition 11
- To inscribe an equilateral and equiangular pentagon in a given circle.
- Proposition 12
- To circumscribe an equilateral and equiangular pentagon about a given circle.
- Proposition 13
- To inscribe a circle in a given equilateral and equiangular pentagon.
- Proposition 14
- To circumscribe a circle about a given equilateral and equiangular pentagon.
- Proposition 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 we can both
inscribe a circle in a given hexagon and circumscribe one about it.

- Proposition 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 both inscribe
a circle in the given fifteen-angled figure and circumscribe one about it.

Euclid's Elements: Book III <--- Book IV ---> Euclid's Elements: Book V