Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/226

202 For the same reason the triangle ECD is to the triangle LHK in the duplicate ratio of EC to LH.

Therefore the triangle EBC is to the triangle LGH as the triangle ECD is to the triangle LHK. [V. 11.

But it has been shewn that the triangle EBC is to the triangle LGH as the triangle ABE is to the triangle FGL.

Therefore as the triangle ABE is to the triangle FGL, so is the triangle EBC to the triangle LGH, and the triangle ECD to the triangle LHK; [V. 11.

and therefore as one of the antecedents is to its consequent so are all the antecedents to all the consequents; [V. 12.

that is, as the triangle ABE is to the triangle FGL so is the polygon ABCDE to the polygon FGHKL.

But the triangle ABE is to the triangle FGL in the duplicate ratio of the side AB to the homologous side FG; [VI. 19.

therefore the polygon ABCDE is to the polygon FGHKL in the duplicate ratio of the side AB to the homologous side FG.

Wherefore, similar polygons &c.

1. In like manner it may be shewn that similar four- sided figures, or figures of any number of sides, are to one another in the duplicate ratio of their homologous sides; and it has already been shewn for triangles; therefore universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.

2. If to AB and FG, two of the homologous sides, a third proportional M be taken, [VI. 11.

then AB has to M the duplicate ratio of that which AB has to FG. [V. Definition 10.