If a straight line through the centre of a circle bisect a chord, then it is perpendicular to the chord, and if it be perpendicular to the chord it bisects it.
An easy consequence of this is the following theorem, which is essentially the same as Prop. 4:—
Two chords of a circle, of which neither passes through the centre, cannot bisect each other.
These last three theorems are fundamental for the theory of the circle. It is to be remarked that Euclid never proves that a straight line cannot have more than two points in common with a circumference.
§ 29. The next two propositions (5 and 6) might be replaced by a single and a simpler theorem, viz:—
Two circles which have a common centre, and whose circumferences have one point in common, coincide.
Or, more in agreement with Euclid’s form:—
Two different circles, whose circumferences have a point in common, cannot have the same centre.
That Euclid treats of two cases is characteristic of Greek mathematics.
The next two propositions (7 and 8) again belong together. They may be combined thus:—