Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/239

Rh Then, because in the triangles BGC,CGK the two sides BG, GC are equal to the two sides CG, GK, each to each; and that they contain equal angles; [III. 27. therefore the base BC is equal to the base CK, and the triangle BGC is equal to the triangle CGK, [I. 4.

And because the arc BC is equal to the arc CK, [Constr. the remaining part when BC is taken from the circumference is equal to the remaining part when CK is taken from the circumference; therefore the angle BXC is equal to the angle COK. [III. 27. Therefore the segment BXC is similar to the segment COK; [III. Definition 11. and they are on equal straight lines BC, CK. But similar segments of circles on equal straight lines are equal to one another; [III. 24. therefore the segment BXC is equal to the segment COK.

And the triangle BGC was shewn to be equal to the triangle CGK; therefore the whole, the sector BGC, is equal to the whole, the sector CGK [Axiom 2. For the same reason the sector KGL is equal to each of the sectors BGC, CGK,

In the same manner the sectors EHF, FHM, MHN may be shewn to be equal to one another.

Therefore whatever multiple the arc BL is of the arc BC, the same multiple is the sector BGL of the sector BGC;