Because Triangles DGK, GHM, HEN are on equal bases and have their base-∠s respectively equal,
∴ DK, GM, HN are equal. [Euc. I. 26.
Join GL.
Because DL, GM are equally inclined to DE,
∴ they are equally inclined to GL; [(μ).
∴ ∠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. [I. 26.
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. [I. 4.
Because GK, EF are equally inclined to DE,
∴ they are equally inclined to DF; [(μ).
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.
