lines. PROPOSITION XXV. PROBLEM. 306. To find a mean proportional between two given Let the two given lines be m and n. It is required to find a mean proportional between m and n. On the straight line A E take A Cm, and C B = n. On A B as a diameter describe a semi-circumference. At C erect the 1 CH. CH is a mean proportional between m and n. Draw HB and HA. The AHB is a rt. ≤, (being inscribed in a semicircle), $ 204 and HC is a let fall from the vertex of a rt. to the hypotenuse. .. AC CH :: CH: CB, $ 289 (the let fall from the vertex of the rt. to the hypotenuse is a mean proportional between the segments of the hypotenuse). Substitute for A C and C B their equals m and n. Then m: CH :: CH : п. Q. E. F. 307. COROLLARY. If from a point in the circumference a perpendicular be drawn to the diameter, and chords from the point to the extremities of the diameter, the perpendicular is a mean proportional between the segments of the diameter, and each chord is a mean proportional between its adjacent segment and the diameter. 308. To divide one side of a triangle into two parts proportional to the other two sides. It is required to divide the side BC into two such parts that the ratio of these two parts shall equal the ratio of the other two sides, A C and A B. Produce CA to F, making A F = A B. (a line drawn through two sides of a ▲ to the third side divides those sides 309. COROLLARY. The line A E bisects the angle C A B. 310. DEF. A straight line is said to be divided in extreme and mean ratio, when the whole line is to the greater segment as the greater segment is to the less. PROPOSITION XXVII. PROBLEM. 311. To divide a given line in extreme and mean ratio. It is required to divide A B in extreme and mean ratio. At B erect a BC, equal to one-half of A B. From C as a centre, with a radius equal to C B, describe a O. Since AB is to the radius CB at its extremity, it is tangent to the circle. Through C draw A D, meeting the circumference in E and D. On A B take A H= A E. H is the division point of A B required. For AD : A B A B AE, § 292 (if from a point without the circumference a secant and a tangent be drawn, the tangent is a mean proportional between the whole secant and the part without the circumference). Then AD - AB: A B :: A B - AE: A E. § 265 AH :: AH: H B. $263 Whence, by inversion, A B Q. E. F. REMARK. AB is said to be divided at H, internally, in extreme and mean ratio. If BA be produced to H', making A H' equal to AD, A B is said to be divided at H', externally, in extreme and mean ratio. When a line is divided internally and externally in the same ratio, it is said to be divided harmonically. Thus AB A C B + + Dis divided harmoni cally at C and D, if CA: C B:: DA: DB; that is, if the ratio of the distances of C from A and B is equal to the ratio of the distances of D from A and B. This proportion taken by alternation gives : AC:AD::BC: BD; that is, CD is divided harmonically at the points B and A. The four points A, B, C, D, are called harmonic points; and the two pairs A, B, and C, D, are called conjugate points. Ex. 1. To divide a given line harmonically in a given ratio. 2. To find the locus of all the points whose distances from two given points are in a given ratio. 312. Upon a given line homologous to a given side of a given polygon, to construct a polygon similar to the given polygon. B E Let A' E' be the given line, homologous to A E of the given polygon ABCDE. It is required to construct on A' E' a polygon similar to the given polygon. From E draw the diagonals EB and EC. From E' draw E' B', making A' E' BLAE B. Also from A' draw A' B', making B'A' E'Z BAE, and meeting E' B' at B'. = The two A A B E and A' B' E' are similar, § 280 (two are similar if they have two of the one equal respectively to two of the other). Also from E' draw E' C', making ▲ B'E' C' From B' draw B'C', making ▲ E' B'C' = ZE BC, and meeting E' C' at C'. Then the two A EBC and E' B'C' are similar, $280 (two are similar if they have two of the one equal respectively to two of the other). In like manner construct A E'C' D' similar to ▲ ECD. Then the two polygons are similar, $293 (two polygons composed of the same number of similar to each other and similarly placed, are similar). .. A' B' C' D' E' is the required polygon. Q. E. F. |