§ 31. Propositions 14 and 15 relate to the length of chords. The first says that equal chords are equidistant from the centre, and that chords which are equidistant from the centre are equal;
Whilst Prop. 15 compares unequal chords, viz. Of all chords the diameter is the greatest, and of other chords that is the greater which is nearer to the centre; and conversely, the greater chord is nearer to the centre.
§ 32. In Prop. 16 the tangent to a circle is for the first time introduced. The proposition is meant to show that the straight line at the end point of the diameter and at right angles to it is a tangent. The proposition itself does not state this. It runs thus:—
Prop. 16. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle.
Corollary.—The straight line at right angles to a diameter drawn through the end point of it touches the circle.
The statement of the proposition and its whole treatment show the difficulties which the tangents presented to Euclid.
Prop. 17 solves the problem through a given point, either in the circumference or without it, to draw a tangent to a given circle.
Closely connected with Prop. 16 are Props. 18 and 19, which state (Prop. 18), that the line joining the centre of a circle to the point of contact of a tangent is perpendicular to the tangent; and conversely (Prop. 19), that the straight line through the point of contact of, and perpendicular to, a tangent to a circle passes through the centre of the circle.
§ 33. The rest of the book relates to angles connected with a circle, viz. angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at the centre and angles at the circumference. Between these two kinds of angles exists the important relation expressed as follows:—
Prop. 20. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.