Upon a given straight line to describe a rectilineal figure similar to a given rectilineal figure, and similarly situated. Similar triangles are to one another in the duplicate ratio of their homologous sides. COR.-Hence it is manifest that if three straight lines be proportional, as the first is to the third, so is any triangle upon the first, to a similar and similarly described triangle upon the second. PROP. XX. THEOR. Similar polygons may be divided into the same number of similar triangles, each similar pair of which are proportional to the polygons; and the polygons are to each other in the duplicate ratio of their homologous sides. COR. 1.-It is obvious that what is here demonstrated of similar polygons may be in like manner proved of similar quadrilateral figures, which are therefore to one another in the duplicate ratio of their homologous sides. COR. 2.-If, to AB, FG homologous sides of two similar figures, a third proportional M be taken, it is plain that the figure on AB is to a similar and similarly placed figure on FG, as AB is to M. PROP. XXI. THEOR. Rectilinear figures which are similar to the same rectilinear figure are also similar to one another. PROP. XXII. THEOR. If four straight lines be proportional, the similar rectilinear figures similarly described upon them, shall also be proportional; and if similar rectilinear figures similarly described upon four straight lines be proportional, those straight lines shall be proportional. |