Book 4
DEFINITIONS.
1
1
A rectilineal figure is said to be inscribed in a rectilineal figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.
2
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
3
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.
4
4
A rectilineal figure is said to be circumscribed about a circle, when each side of the circumscribed figure touches the circumference of the circle.
5
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
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
7
A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.
BOOK IV. PROPOSITIONS
PROPOSITION 1.
1
Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle.
1
Let ABC be the given circle, and D the given straight line not greater than the diameter of the circle; thus it is required to fit into the circle ABC a straight line equal to the straight line D.
1
Let a diameter BC of the circle ABC be drawn.
1
Then, if BC is equal to D, that which was enjoined will have been done; for BC has been fitted into the circle ABC equal to the straight line D.
1
But, if BC is greater than D, let CE be made equal to D, and with centre C and distance CE let the circle EAF be described; let CA be joined.
1
Then, since the point C is the centre of the circle EAF, CA is equal to CE.
1
But CE is equal to D; therefore D is also equal to CA.
