Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/125

Rh PROPOSITION 29. THEOREM.

In equal circles, equal arcs are subdtended by equal straight lines.

Let ABC, DEF be equal circles, and let BGG, EHF be equal arcs in them, and join BC, EF: the straight line BC shall be equal to the straight line EF.

Take K, L, the centres of the circles, [III. 1. and join BK, KG, EL, LF. Then, because the arc BCG is equal to the arc EHF, [Hypothesis. the angle BKC is equal to the angle ELF. [III. 27. And because the circles ABC, DEF are equal, [Hypothesis. the straight lines from their centres are equal; [III. Def. 1. therefore the two sides BK, KC are equal to the two sides EL, LF, each to each;

and they contain equal angles; therefore the base BC is equal to the base EF. [I. 4.

Wherefore, in equal circles &c.

PROPOSITION 30. PROBLEM.

To bisect a given arc, that is, to divide it into two equal parts.