Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/97

Rh Draw within it any straight line AB and bisect AB at D; [I. 10. from the point D draw DC at right angles to AB; [I. 11. produce CD to meet the circumference at E, and bisect CE at F. [I. 10. The point F shall be the centre of the circle ABC.

For if F be not the centre, if possible, let G be the centre; and join GA, GD, GB. Then, because DA is equal to DB, [Construction. and DG is common to the two triangles ADG, BDG; the two sides AD, DG are equal to the two sides BD, DG, each to each; and the base GA is equal to the base GB, because they are drawn from the centre G; [I. Definition 15. therefore the angle ADG is equal to the angle BDG. [I. 8. But when a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle; [I. Definition 10. therefore the angle BDG is a right angle. But the angle BDF is also a right angle. [Construction.

Therefore the angle BDG is equal to the angle BDF, [Ax.ll the less to the greater; which is impossible. Therefore G is not the centre of the circle ABC.

In the same manner it may be shewn that no other point out of the line CE is the centre; and since CE is bisected at F, any other point in CE divides it into unequal parts, and cannot be the centre. Therefore no point but F is the centre; that is, F is the centre of the circle ABC: which was to he found.

. From this it is manifest, that if in a circle a straight line bisect another at right angles, the centre of the circle is in the straight line which bisects the other.