The first of the figures thus constructed is said to be directly similar, and the second inversely similar to the given figure. These technical terms are due to Hamilton: see “Elements of Quaternions,” page 112.
Cor. 1.—Twice as many polygons may be constructed on AB similar to a given polygon CDEFG as that figure has sides.
Cor. 2.—If the figure ABHIJ be applied to CDEFG so that the point A will coincide with C, and that the line AB may be placed along CD, then the points H, I, J will be respectively on the lines CE, CF, CG; also the sides BH, HI, IJ of the one polygon will be respectively parallel to their homologous sides DE, EF, FG of the other.
Cor. 3.—If lines drawn from any point O in the plane of a figure to all its angular points be divided in the same ratio, the lines joining the points of division will form a new figure similar to, and having every side parallel to, the homologous side of the original.
PROP. XIX.—Theorem.
Similar triangles (ABC, DEF) have their areas to one another in the duplicate ratio of their homologous sides.
Dem.—Take BG a third proportional to BC, EF [xi.]. Join AG. Then because the triangles ABC, DEF are similar, AB : BC :: DE : EF; hence (alternately) AB : DE :: BC : EF; but BC : EF :: EF : BG (const.); therefore [V. xi.] AB : DE :: EF : BG; hence the sides of the triangles ABG, DEF about the equal angles B, E are reciprocally proportional; therefore the triangles are equal. Again, since the lines BC, EF, BG are continual proportionals, BC : BG in the duplicate ratio of BC : EF [V. Def. x.]; but BC : BG :: triangle ABC : ABG. Therefore ABC : ABG in the duplicate ratio of BC : EF; but it has been proved that the triangle ABG is equal to DEF. Therefore the triangle ABC is to the triangle DEF in the duplicate ratio of BC : EF.
This is the first Proposition in Euclid in which the technical term “duplicate ratio” occurs. My experience with pupils is, that they find it very difficult to understand either Euclid’s proof or his definition. On this account I submit the following alternative proof, which, however, makes use of a new definition of the duplicate ratio of two lines, viz. the ratio of the squares (see Annotations on V. Def. x.) described on these lines.
On AB and DE describe squares, and through C and F draw lines parallel to AB and DE, and complete the rectangles AI, DN.
Now, the triangles JAC, ODF are evidently equiangular.