Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/122

98 PROPOSITION 26. THEOREM.

In equal circles, equal angles stand on equal arcs, whether they be at the centres or circumferences.

Let ABC, DEF be equal circles; and let BGC, EHF be equal angles in them at their centres, and BAC, EDF equal angles at their circumferences: the arc BKC shall be equal to the arc ELF.

Join BC, EF.

Then, because the circles ABC, DEF are equal, [Hyp. the straight lines from their centres arc equal; [III. Def. 1. therefore the two sides BG, GC are equal to the two sides EH, HF, each to each; and the angle at G is equal to the angle at H; [Hypothesis. therefore the base BC is equal to the base EF. [I. 4.

And because the angle at A is equal to the angle at D,[Hyp. the segment BAC dissimilar to the segment EDF; [III. Def. ll. and they are on equal straight lines BC, EF. But similar segments of circles on equal straight lines are equal to one another; [III. 24. therefore the segment BAC be equal to the segment EDF.

But the whole circle ABC is equal to the whole circle DEF; [Hypothesis. therefore the remaining segment BKC is equal to the remaining segment ELF; [Axiom 3. therefore the arc BKC is equal to the arc ELF.

Wherefore, in equal circies &c.