PROPOSITION XXIX. PROBLEM. 356. To divide a given line in extreme and mean Let AB be the given line. To divide AB in extreme and mean ratio. Construction. At B erect a 1 BE equal to one-half of AB. From E as a centre, with a radius equal to EB, describe a O. Draw AE, meeting the circumference in Fand G. Then AB is divided internally at C and externally at C" in extreme and mean ratio. (if from a point without a O a secant and a tangent are drawn, the tangent is a mean proportional between the whole secant and the external segment). Then by § 301 and § 300, AG-AB: AB= AB-AF: AF, AG+AB: AG AB+ AF AB. = (1) (2) PROPOSITION XXX. PROBLEM. 357. Upon a given line homologous to a given side of a given polygon, to construct a polygon similar to the given polygon. Let A'E' be the given line homologous to AE of the given polygon ABCDE. To construct on A'E' a polygon similar to the given polygon. Construction. From E draw the diagonals EB and EC. From E' draw E'B', E'C', and E'D', making A'E'B', B'E'C', and C'E'D' equal respectively to AEB, BEC, and CED. From A' draw A'B', making ▲ E'A'B'= Z EAB, From B' draw B'C', making ▲ E'B'C' = ▲ EBC, From C' draw C'D', making ▲ E'C'D' = ▲ ECD, Then A'B'C'D'E' is the required polygon. Proof. The corresponding A ABE and A'B'E', EBC and E'B'C', ECD and E'C'D' are similar, (two are similar if they have two § 322 of the one equal respectively to two of the other). Then the two polygons are similar, § 331 (two polygons composed of the same number of similar to each other and similarly placed, are similar). Q. E. F. PROBLEMS OF COMPUTATION. 219. To compute the altitudes of a triangle in terms of its sides. At least one of the angles A or B is acute. Suppose it is the angle B. 220. To compute the medians of a triangle in terms of its sides. 221. To compute the bisectors of a triangle in terms of the sides. 222. To compute the radius of the circle circumscribed about a triangle in terms of the sides of the triangle. By 350, AB × AC = AE× AD, A с 223. If the sides of a triangle are 3, 4, and 5, what kind of an angle is opposite 5? 224. If the sides of a triangle are 7, 9, and 12, what kind of an angle is opposite 12? 225. If the sides of a triangle are 7, 9, and 11, what kind of an angle is opposite 11? 226. The legs of a right triangle are 8 inches and 12 inches; find the lengths of the projections of these legs upon the hypotenuse, and the distance of the vertex of the right angle from the hypotenuse. 227. If the sides of a triangle are 6 inches, 9 inches, and 12 inches, find the lengths (1) of the altitudes; (2) of the medians; (3) of the bisectors; (4) of the radius of the circumscribed circle. THEOREMS. 228. Any two altitudes of a triangle are inversely proportional to the corresponding bases. 229. Two circles touch at P. Through P three lines are drawn, meeting one circle in A, B, C, and the other in A', B', C', respectively. Prove that the triangles ABC, A'B'C' are similar. 230. Two chords AB, CD intersect at M, and A is the middle point of the arc CD. Prove that the product AB× AM remains the same if the chord AB is made to turn about the fixed point A. HINT. Draw the diameter AE, join BE, and compare the triangles thus formed. 231. The sum of the squares of the segments of two perpendicular chords is equal to the square of the diameter of the circle. If AB, CD are the chords, draw the diameter BE, join AC, ED, BD, and prove that AC-ED. Apply ? 338. 232. In a parallelogram ABCD, a line DE is drawn, meeting the diagonal AC in F, the side BC in G, and the side AB produced in E. Prove that D2 = FG × FE. 233. The tangents to two intersecting circles drawn from any point in their common chord produced, are equal. (8 348.) 234. The common chord of two intersecting circles, if produced, will bisect their common tangents, ( 348.) 235. If two circles touch each other, their common tangent is a mean proportional between their diameters. HINT. Let AB be the common tangent. Draw the diameters AC, BD. Join the point of contact P to A, B, C, and D. Show that APD and BPC are straight lines to each other, and compare ▲ ABC, ABD. 236. If three circles intersect one another, the common chords all pass through the same point. HINT. Let two of the chords AB and CD meet at O. Join the point of intersection E to O, and suppose that EO produced meets the same two circles at two different points P and Q. Then prove that OP=OQ; hence, that the points P and Q coincide. PO D |