65. Is the greatest angle of a triangle whose sides are 8, 9, and 12, acute, right, or obtuse? 66. Is the greatest angle of a triangle whose sides are 12, 35, and 37, acute, right, or obtuse? 67. If two adjacent sides and one of the diagonals of a parallelogram are 7, 9, and 8, respectively, find the other diagonal. (One-half of either diagonal is a median of the ▲ whose sides are, respectively, the given sides and the other diagonal of the □.) 68. If D is the intersection of the perpendiculars from the vertices of triangle ABC to the opposite sides, prove 69. If a parallel to hypotenuse AB of right triangle ABC meets AC and BC at D and E, respectively, prove AE2 + BD2 = AB2 + DE2. 70. The diameters of two circles are 12 and 28, respectively, and the distance between their centres is 29. Find the length of the common tangent which cuts the straight line joining the centres. (Find the drawn from the centre of the smaller O to the radius of the greater O produced through the point of contact.) 71. State and prove the converse of Prop. XXIII., III. (Fig. of Prop. XXIII. A ABC and ACD are similar.) 72. State and prove the converse of Prop. XXIII., II. 73. The sum of the squares of the distances of B any point in the circumference of a circle from the vertices of an inscribed square, is equal to twice the square of the diameter of the circle. (§ 195.) (To prove PA2 +PB2 +PC2 + PD2=2 AC2.) Α 74. The sides AB, BC, and CA, of triangle ABC, are 13, 14, and 15, respectively. Find the segments into which AB and BC are divided by perpendiculars drawn from C and A, respectively. (BAC and ACB are acute by § 98. Find the segments by § 277.) 75. In right triangle ABC is inscribed a square DEFG, having its vertices D and G in hypotenuse BC, and its vertices E and F in sides AB and AC, respectively. Prove BD: DE = DE : CG. (Prove & BDE and CFG similar.) Note. For additional exercises on Book III., see p. 226. CONSTRUCTIONS. PROP. XXXII. Problem. 289. To divide a given straight line into any number of equal parts. Required to divide AB into four equal parts. Construction. On the indefinite line AC, take any convenient length AD; on DC take_DE = AD; on EC take EF= AD; on FC take FG = AD; and draw line BG. Draw lines DH, EK, and FL || BG, meeting AB at H, K, and L, respectively. :. AH = HK = KL = LB. (§ 242) PROP. XXXIII. PROBLEM. 290. To construct a fourth proportional (§ 231) to three given straight lines. Required to construct a fourth proportional to m, n, and p. Construction. Draw the indefinite lines AB and AC, making any convenient with each other. On AB take AD = m; on DB take DE AFP. =n; on AC take Draw line DF; also, line EG || DF, meeting AC at G. Then, FG is a fourth proportional to m, n, and p. Proof. Since DF is to side EG of ▲ AEG, 291. Cor. If we take AF = n, the proportion becomes m: n=n: FG. (?) In this case, FG is a third proportional (§ 230) to m and n. PROP. XXXIV. PROBLEM. 292. To construct a mean proportional (§ 230) between two given straight lines. n m m B n Given lines m and n. Required to construct a mean proportional between m and n. Construction. On the indefinite line AE, take AB = m; on BE take BC= n. With AC as a diameter, describe the semi-circumference ADC. Draw line BDLAC, meeting the arc at D. Then, BD is a mean proportional between m and n. (The proof is left to the pupil; see § 270.) 293. Sch. By aid of § 292, a line may be constructed equal to Va, where a is any number whatever. Thus, to construct a line equal to √3, we take AB equal to 3 units, and BC equal to 1 unit. Then, BD =√AB × BC (§ 232) = √3 × 1 = √3. PROP. XXXV. PROBLEM. 294. To divide a given straight line into parts proportional to any number of given lines. Given line AB, and lines m, n, and p. Required to divide AB into parts proportional to m, n, and p. Construction. On the indefinite line AC, take AD=m; on DC take DE = n; on EC take EF = p; and draw line BF. Draw lines DG and EH || to BF, meeting AB at G and H, respectively. Then, AB is divided into parts AG, GH, and HB proportional to m, n, and p, respectively. Proof. Since DG is to side. EH of ▲ AEH, And since EH is to side BF of ▲ ABF, Ex. 76. Construct a line equal to √2; to √5; to √6. (?) (1) (2) (?) PROP. XXXVI. PROBLEM. 295. Upon a given side, homologous to a given side of a given polygon, to construct a polygon similar to the given polygon. Given polygon ABCDE, and line A'B'. Required to construct upon side A'B', homologous to AB, a polygon similar to ABCDE. Construction. Divide polygon ABCDE into A by drawing diagonals EB and EC. At A' construct ▲ B'A'E' = ZA; and draw line B'E', · making A'B'E' = LABE, meeting A'E' at E'. Then, ▲ A'B'E' will be similar to ▲ ABE. (?) In like manner, construct ▲ B'C'E' similar to ▲ BCE, and AC'D'E' similar to ▲ CDE. Then, polygon A'B'C'D'E' will be similar to polygon ABCDE. (§ 266) 296. Def. A straight line is said to be divided by a given point in extreme and mean ratio when one of the segments (§ 250) is a mean proportional between the whole line and the other segment. Thus, line AB is divided internally in extreme and mean ratio at C if and externally in extreme and mean ratio at D if |