...similar to AD E. The polygon FGHIK will be similar to ABCDE, as required. For these two polygons arc composed of the same number of triangles, similar each to each, and similarly situated (Theo. XX. Cor., Bk. IV.). PROBLEM • VII. To inscribe a square in a given circle. Draw two diameters,...

...triangle FIK similar to AD E. The polygon FG HIK will be similar to ABCDE, as required. For these two polygons are composed of the same number of triangles, similar each to each, and similarly situated (Theo. XX. Cor., Bk. IV.). PROBLEM VII. To inscribe a square in a given circle. Draw two diameters,...

...number of triangles, which are similar each to each, and similarly situated. Cor. Conversely, if two polygons are composed of the same number of triangles, similar each to each, and similarly situated, the polygons are similar. For, because the triangles are similar, the angle ABC is equal to FGH ; and...

...be shown that the remaining triangles are similar each to each. XXVIII. Theorem. Conversely, if tu'o polygons are composed of the same number of triangles similar each to each, and similarly situated, the polygons are similar. . HYPOTII. In the two polygons, ABCDE and abcde, A ABC ~ abc, ACD ~ acd,...

...equal to ADxDC (Prop. XIX.); and, by substitution, ABxBC=ADxDC+BD 2 . PROPOSITION XXIII.—THEOREM. Two similar polygons are composed of the same number of triangles, similar to each other, and similarly situated* Let ABCDE and FGKLM be two similar polygons; from any angle,...

...triangles be shown to be similar, whatever were the number of sides of the proposed polygons ; therefore, two similar polygons are composed of the same number of triangles, similar and similarly situated. SCHOLIUM. The converse proposition is equally true : If two polygons are composed...

...:: HK : FG :: CK : CG ELEMENTS OF PLANE GEOMETRY. THEOREM XX. 277. Two polygons are similar if they are composed of the same number of triangles similar each to each and similarly placed. Let the As ABE, EBD, and DBC, of which the polygon ABСDE is composed, be respectively similar...

...is similar to AB CD E. Hence we arrive at the following useful theorem : — Theorem I. — If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar. The converse theorem is also true. State it. If each side of a polygon...

...PROPOSITION V. Two triangles are similar when their homologous sides are proportional. PROPOSITION VI. If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar. PROPOSITION VII. Two similar polygons may be decomposed into the...

...coincides with F. :. EF prolonged passes through O. SIMILAR POLYGONS. PROPOSITION XII. THEOREM. 331. If two polygons are composed of the same number of triangles, similar each to each, and similarly placed, the polygons are similar. E BCW l!* In the two polygons ABCDK and A'B ' <yD>E', let the triangles...