...base BC is divided at the points F, G, H. THEOREM XII. Two similar polygons can fee decomposed into the same number of triangles, similar each to each, and similarly placed. Let there be two similar polygons ABCDE, A' B' C' ' D' E'. From their homologous summits draw the homologous...

...by the preceding theorem, we have AP=AC ab ac But, from the hypothesis [1J, we have by (12), AC AB ab ' ac whence, AP = AB, which is impossible unless...the triangles ABC, A CD, etc.; and let the polygon A'B'C'D', etc., be composed of the triangles A'B'C', A'C'D', etc., similar to ABC, ACD, etc., respectively,...

...same 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...

...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...

...(12), ab ac PROPOSITION X.—THEOREM. 38. If two polygons are composed of the same number of triangle* similar each to each and similarly placed, the polygons are similar. Let the polygon ABCD, etc., be composed of the triangles ABC, A CD, etc.; and let the polygon A'B'C'D', etc., be composed...

...polygon = 180° X the number of sides less tw0. 4. Two polygons are equal, if they can be divided into the same number of triangles, similar each to each, and similarly placed. 5. A polygon is equilateral, if its sides are equal ; equiangular, if its angles are equal ; regular,...

...of triangles, similar and similarly situated. SCHOLIUM. The converse proposition is equally true : If two polygons are composed of the same number of triangles, similar and similarly situated, those two polygons will be similar. For the similarity of the respective triangles...

...:: 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 to the...

...similar. (237) QED ELEMENTS OF PLANE GEOMETRY. THEOREM XXI. 278. Two similar polygons can be divided into the same number of triangles similar each to each and similarly placed. Let the polygon ABCDE be similar to the polygon FGHKM, and let diagonals be drawn from the vertices...

...area of the first than that of the second? An*. 2$. THEOREM IX. Similar polygons may be resolved into the same number of triangles, similar, each to each, and similarly placed. Let P and P' be two similar polygons; then, can they be resolved into the same number of similar triangles,...