For the same reason we shall call a straight line joining two points on the circumference of a circle a “chord.”
Definitions 4 and 5 may be replaced with a slight generalization by the following:—
Definition.—By the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line.
§ 27. From the definition of a circle it follows that every circle has a centre. Prop. 1 requires to find it when the circle is given, i.e. when its circumference is drawn.
To solve this problem a chord is drawn (that is, any two points in the circumference are joined), and through the point where this is bisected a perpendicular to it is erected. Euclid then proves, first, that no point off this perpendicular can be the centre, hence that the centre must lie in this line; and, secondly, that of the points on the perpendicular one only can be the centre, viz. the one which bisects the parts of the perpendicular bounded by the circle. In the second part Euclid silently assumes that the perpendicular there used does cut the circumference in two, and only in two points. The proof therefore is incomplete. The proof of the first part, however, is exact. By drawing two non-parallel chords, and the perpendiculars which bisect them, the centre will be found as the point where these perpendiculars intersect.
§ 28. In Prop. 2 it is proved that a chord of a circle lies altogether within the circle.
What we have called the first part of Euclid’s solution of Prop. 1 may be stated as a theorem:—
Every straight line which bisects a chord, and is at right angles to it, passes through the centre of the circle.
The converse to this gives Prop. 3, which may be stated thus:—