If from a point in a plane of a circle, which is not the centre, straight lines be drawn to the different points of the circumference, then of all these lines one is the shortest, and one the longest, and these lie both in that straight line which joins the given point to the centre. Of all the remaining lines each is equal to one and only one other, and these equal lines lie on opposite sides of the shortest or longest, and make equal angles with them.
Euclid distinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circumference.
From the last proposition it follows that if from a point more than two equal straight lines can be drawn to the circumference, this point must be the centre. This is Prop. 9.
As a consequence of this we get
If the circumferences of the two circles have three points in common they coincide.
For in this case the two circles have a common centre, because from the centre of the one three equal lines can be drawn to points on the circumference of the other. But two circles which have a common centre, and whose circumferences have a point in common, coincide. (Compare above statement of Props. 5 and 6.)
This theorem may also be stated thus:—
Through three points only one circumference may be drawn; or, Three points determine a circle.
Euclid does not give the theorem in this form. He proves, however, that the two circles cannot cut another in more than two points (Prop. 10), and that two circles cannot touch one another in more points than one (Prop. 13).
§ 30. Propositions 11 and 12 assert that if two circles touch, then the point of contact lies on the line joining their centres. This gives two propositions, because the circles may touch either internally or externally.