Page:Carroll - Euclid and His Modern Rivals.djvu/298

260

Because Triangles DGK, GHM, HEN are on equal bases and have their base-∠s respectively equal,

∴ DK, GM, HN are equal.

Join GL.

Because DL, GM are equally inclined to DE,

∴ they are equally inclined to GL;

∴ ∠s KLG, LGM are equal.

Similarly, ∵ GK, HL are equally inclined to DE,

∴ they are equally inclined to GL;

∴ ∠s KGL, GLM are equal.

Because Triangles LGK, GLM are on same base LG and have their base-∠s respectively equal,

∴ KL = GM, i.e. = DK.

Similarly it may be proved that LF = HN, i.e. = DK.

Hence DE, DF are equimultiples of DG, DK, i.e. of AB, DK,

but they are also equimultiples of AB, AC;

∴ DK = AC.

Because Triangles ABC, DGK have ∠s A, D equal, and AB, AC respectively equal to DG, DK,

∴ ∠s, B, DGK are equal, and likewise ∠s C, DKG.

Because GK, EF are equally inclined to DE,

∴ they are equally inclined to DF;

i.e. ∠s DKG, DFE are equal;

∴ ∠s B, C are respectively equal to ∠s E, F.

Hence, two Triangles, which have their vertical angles equal, and the 2 sides of the one respectively equimultiples of those of the other, have their base-angles respectively equal.