39. If two sides of a triangle be given in position, and if their included angle be equal to an angle of an equilateral triangle, the locus of the centre of its nine-points circle is a right line. 40. If, in the hypothesis and notation of Ex. 34, α, β denote any two suffixes whose sum is less than 8, and of which α is the greater, For instance, ρ1ρ4 = R(ρ3 + ρ5) [III. xxxv., Ex. 7]. In the same case, if the suffixes be greater than 8, For instance, ρ8ρ2 = R(ρ6 − ρ7) [III. xxxv., Ex. 6]. 41. Two lines are given in position: draw a transversal through a given point, forming with the given lines a triangle of given perimeter. 42. Given the vertical angle and perimeter of a triangle, construct it with either of the following data: 1. The bisector of the vertical angle; 2. the perpendicular from the vertical angle on the base; 3. the radius of the inscribed circle. 43. In a given circle inscribe a triangle so that two sides may pass through two given points, and that the third side may be a maximum or a minimum. 44. If s be the semiperimeter of a triangle, r′, r′′, r′′′, the radii of its escribed circles, 45. The feet of the perpendiculars from the extremities of the base on either bisector of the vertical angle, the middle point of the base, and the foot of the perpendicular from the vertical angle on the base, are concyclic. 46. Given the base of a triangle and the vertical angle; find the locus of the centre of the circle passing through the centres of the escribed circles. 47. The perpendiculars from the centres of the escribed circles of a triangle on the corresponding sides are concurrent. 48. If AB be the diameter of a circle, and PQ any chord cutting AB in O, and if the lines AP, AQ intersect the perpendicular to AB at O, in D and E respectively, the points A, B, D, E are concyclic. 49. If the sides of a triangle be in arithmetical progression, and if R, r be the radii of the circumscribed and inscribed circles; then 6Rr is equal to the rectangle contained by the greatest and least sides. 50. Inscribe in a given circle a triangle having its three sides parallel to three given lines. 51. If the sides AB, BC, &c., of a regular pentagon be bisected in the points A′, B′, C′, D′, E′, and if the two pairs of alternate sides, BC, AE; AB, DE, meet in the points A′′, E′′, respectively, prove 52. In a circle, prove that an equilateral inscribed polygon is regular, and also an equilateral circumscribed polygon, if the number of sides be odd. 53. Prove also that an equiangular circumscribed polygon is regular, and an equiangular inscribed polygon, if the number of sides be odd. 54. The sum of the perpendiculars drawn to the sides of an equiangular polygon from any point inside the figure is constant. 55. Express the sides of a triangle in terms of the radii of its escribed circles. BOOK V. THEORY OF PROPORTION ________________ DEFINITIONS.