Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/98

74 PROPOSITION 2. THEOREM.

If any two points he taken in the circumference of a circle, the straight line which joins them shall fall within the circle.

Let ABC be a circle, and A and B any two points in the circumference: the straight line drawn from A to B shall fall within the circle.

For if it do not, let it fall, if possible, without, as AEB. Find D the centre of the circle ABC; [III. 1. and join DA, DB; in the arc AB take any point F, join DF, and produce it to meet the straight line AB at E. Then, because DA is equal to DB, [I. Definition 15. the angle DAB is equal to the angle DBA. [I. 5. And because AE, a side of the triangle DAE, is produced to B, the exterior angle DEB is greater than the interior opposite angle DAE. [I. 16. But the angle DAE was shewn to be equal to the angle DBE; therefore the angle DEB is greater than the angle DBE. But the greater angle is subtended by the greater side; [1. 19. therefore DB is greater than DE. But DB is equal to DF; [I. Definition 15. therefore DF is greater than DE, the less than the greater; which is impossible. Therefore the straight line drawn from A to B does not fall without the circle.

In the same manner it may be shewn that it does not fall on the circumference. Therefore it falls within the circle.

Wherefore, if any two points &c.

PROPOSITION 3. THEOREM. If a straight line drawn through the centre of a circle, bisect a straight line in it which does not pass through the