Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/106

82 PROPOSITION 9. THEOREM.

If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle.

Let the point D be taken within the circle ABC, from which to the circumference there fall more than two equal straight lines, namely DA, DB, DC: the point D shall be the centre of the circle.

For, if not, let E be the centre; join DE and produce it both ways to meet the circumference at F and G; then FG is a diameter of the circle.

Then, because in FG, a diameter of the circle ABC, the point D is taken, which is not the centre, DG is the greatest straight line from D to the circumference, and DC is greater than DB, and DB greater than DA; [III. 7. but they are likewise equal, by hypothesis; which is impossible. Therefore E is not the centre of the circle ABC.

In the same manner it may be shewn that any other point than D is not the centre; therefore D is the centre of the circle ABC.

Wherefore, if a point be taken &c.

PROPOSITION 10. THEOREM.

One circumference of a circle cannot cut another at more than two points.

If it be possible, let the circumference ABC cut the circumference DEF at more than two points, namely, at the points B, G, F.

Take K, the centre of the circle ABC, [II. 1. and join KB, KG, KF.

Then, because K is the centre of the circle ABC,