...lines be cut by parallel lines, they are cut in the same ratio. 2. If, on two bases, there be placed the same number of triangles, similar each to each, and similarly situated, the polygons formed by joining the vertices of the triangles will be similar. 3. A straight line, drawn...

...number of triangles, similar each to each, and similarly situated ; and, conversely, polygons which are composed of the same number of triangles, similar each to each, and similarly situated, are themselves similar. ELEMENTS OP GEOMETRY. while the sides containing these angles are proportional...

...that is, the triangles are similar. PROPOSITION XIX. THEOREM. Similar polygons may be divided into the same number of triangles, similar each to each, and similarly situated ; and, conversely, polygons which are composed of the same number of triangles, similar each to each,...

...remaining triangles are similar, whatever be the number of sides in the polygons proposed : . therefore two similar polygons are composed of the same number of triangles, similar, and similarly situated. Scholium. The converse of the proposition is equally true : If two polygons...

...homologous sides AC, DF, or as the squares of any other two homologous sides. PROPOSITION XXVI. THEOREM. Two similar polygons are composed of the same number...triangles, similar each to each, and similarly situated. Let ABCDE, FGHIK, be two similar polygons. From any angle A, in the polygon ABCDE, ^- „ draw diagonals...

...sides AC, DF, or as the squares of any other two homologous sides. PROPOSITION XXVI. THEOREM. Tico similar polygons are composed of the same number of...triangles, similar each to each, and similarly situated. Let ABCDE, FGHIK, be two similar polygons. From any angle A, in the polygon ABCDE, draw diagonals AC,...

...were parallel to BE. • PROPOSITION XXIII. THEOREM. 104 Let ABCDE, FGHIK, be two similar polygons. From any angle A. in the polygon ABCDE, draw diagonals AC, AD, to the other angles. From A the homologous angle F, in the other polygon FGHIK, draw diagonals FH, FI, to the other angles. These...

...remaining triangles are similar, whatever be the number of sides in the polygons proposed : therefore two similar polygons are composed of the same number of triangles, similar, and similarly situated. FH, Fl to the other angles. For, the similarity of the respective triangles...

...the opposite side, and then taking two-thirds of this line from the vertex. PROPOSITION XIX. THEOREM. Two similar polygons are composed of the same number...triangles similar each to each, and similarly situated. Let ABCDF, GHKLM be two similar polygons. From any angle A in the polygon ABCDF, draw the diagonals...

...triangle FIK similar to AD E. The polygon FGHIK 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 (Prop. XXX. Cor., Bk. IV.). PROBLEM XXXVI. 341. Tioo similar polygons being given, to construct a similar...