350. To construct a polygon similar to two given similar polygons and equivalent to their difference. Let R and R' be two similar polygons, and AB and A'B' two homologous sides. It is required to construct a similar polygon which shall be equivalent to R' - R. From O as a centre, with a radius equal to A' B', describe an arc cutting PX at H. On A" B", homologous to AB, construct the polygon R" similar to R. Then R" is the polygon required. R: R: AB: AB2, § 343 (similar polygons are to each other as the squares on their homologous sides). For PROPOSITION XXI. PROBLEM. 351. To construct a triangle equivalent to a given Let ABC DHE be the given polygon. It is required to construct a triangle equivalent to the given polygon. From D draw D E, and from H draw HF || to D E. Produce AE to meet HF at F, and draw D F. The polygon ABCDF has one side less than the polygon ABCDHE, but the two are equivalent. For the part A B C D E is common, and the A DEF = ADEH, for the base D E is common, and their vertices Fand H are in the line FH to the base, § 325 (A having the same base and equal altitudes are equivalent). Again, draw CF, and draw DK to CF to meet AF produced at K. Draw CK. The polygon ABCK has one side less than the polygon ABCDF, but the two are equivalent. For the part A B C F is common, = and the ACFK ACFD, for the base CF is common, and their vertices K and D are in the line KD || to the base. $325 In like manner we may continue to reduce the number of sides of the polygon until we obtain the ▲ CIK. Q. E. F. PROPOSITION XXII. PROBLEM. 352. To construct a square which shall have a given ratio to a given square. S It is required to construct a square which shall be to R as n is to m. On a straight line take A B =m, and BC = n. On A C as a diameter, describe a semicircle. (being inscribed in a semicircle.) at S, § 204 On SA, or SA produced, take SE equal to a side of R. Then SF is a side of the square required. (a straight line drawn through two sides of a ▲, parallel to the third side, divides those sides proportionally). that is, the square having a side equal to SF will have the same ratio to the square R, as n has to m. Q. E. F. 353. To construct a polygon similar to a given polygon and having a given ratio to it. It is required to construct a polygon similar to R, which shall be to R as n is to m. Find a line, A' B', such that the square constructed upon it shall be to the square constructed upon A B as n is to m. § 352 Upon A'B' as a side homologous to A B, construct the polygon S similar to R. (similar polygons are to each other as the squares on their homologous sides). PROPOSITION XXIV. PROBLEM. 354. To construct a square equivalent to a given paral Let ABCD be a parallelogram, bts base, and a its altitude. It is required to construct a square = ABCD. Upon the line MX take M N = a, and N O = b. Upon MO as a diameter, describe a semicircle. At N erect NPL to MO. Then the square R, constructed upon a line equal to NP, is equivalent to the ABC D. For MN: NP: NP: NO, $ 307 (alet fall from any point of a circumference to the diameter is a mean proportional between the segments of the diameter). .. N p2 = MNX NO = a xb, $ 259 (the product of the means is equal to the product of the extremes). Q. E. F. 355. COROLLARY 1. A square may be constructed equivalent to a triangle, by taking for its side a mean proportional between the base and one-half the altitude of the triangle. 356. COR. 2. A square may be constructed equivalent to any polygon, by first reducing the polygon to an equivalent triangle, and then constructing a square equivalent to the triangle. |