...passes through O. In the same way, Dd is shown to pass through O. PROPOSITION X.— THEOREM. 38. Ij two polygons are composed of the same number of triangles...placed, the polygons are similar. Let the polygon ABCD, etc., be composed of the triangles ABC, ACD, etc.; and let the polygon A'B'C'D', etc., be composed...

...IV.), we have, ABC : DEF : : AC* : DF\ PROPOSITION XXVI. THEOREM. Similar polygons may be divided into the same number of triangles, similar, each to each, and similarly placed. Let ABCDE and FGHIK be two similar poly gone, the angle A being equal to the angle F, B to (?, C to...

...similarity (24). PROPOSITION XI—THEOREM. 39. Conversely, two similar polygons may be decomposed into the same number of triangles similar each to each and similarly placed. Let AB CD, etc., A'B'C'D', etc., be two similar polygons. From two homologous vertices, A and A', let...

...the 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,...

...similar, each to each, to the remaining triangles of the polygon A' B' C' D' E' F' G' H' ; that is, the two polygons are composed of the same number of triangles similar each to each and similarly placed : therefore they are similar to each other [42], WWTBD THEOREM 44. Two parallelograms about the same...

...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...

...BC and EF ; which was to be proved. PROPOSITION XXVI. THEOREM. Similar polygons may be divided into the same number of triangles, similar, each to each, and similarly placed. let ABODE and FGIIIK be two similar polygons, the angle A being equal to the angle F, B to G, C to...

...• and by the preceding theorem, we have ab ac But, from the hypothesis [1J, we have by (12), ab ao whence, AP = AB, which is impossible unless P coincides...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,...

...the 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,...

...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 of the triangles A'B'C', A'C'D', etc., similar to ABC, ACD, etc., respectively, and similarly placed; then, the polygons are...