42. Through two given points describe a circle whose common chord with another given circle may be parallel to a given line, or pass through a given point. 43. Being given the centre of a circle, describe it so as to cut the legs of a given angle along a chord parallel to a given line. 44. If concurrent lines drawn from the angles of a polygon of an odd number of sides divide the opposite sides each into two segments, the product of one set of alternate segments is equal to the product of the other set. 45. If a triangle be described about a circle, the lines from the points of contact of its sides with the circle to the opposite angular points are concurrent. 46. If a triangle be inscribed in a circle, the tangents to the circle at its three angular points meet the three opposite sides at three collinear points. 47. The external bisectors of the angles of a triangle meet the opposite sides in three collinear points. 48. Describe a circle touching a given line at a given point, and cutting a given circle at a given angle. Def.—The centre of mean position of any number of points A, B, C, D, &c., is a point which may be found as follows:—Bisect the line joining any two points A, B, in G. Join G to a third point C; divide GC in H, so that GH = GC. Join H to a fourth point D, and divide HD in K, so that HK = HD, and so on. The last point found will be the centre of mean position of the given points. 49. The centre of mean position of the angular points of a regular polygon is the centre of figure of the polygon. 50. The sum of the perpendiculars let fall from any system of points A, B, C, D, &c., whose number is n on any line L, is equal to n times the perpendicular from the centre of mean position on L. 51. The sum of the squares of lines drawn from any system of points A, B, C, D, &c., to any point P, exceeds the sum of the squares of lines from the same points to their centre of mean position, O, by nOP2. 52. If a point be taken within a triangle, so as to be the centre of mean position of the feet of the perpendiculars drawn from it to the sides of the triangle, the sum of the squares of the perpendiculars is a minimum. 53. Construct a quadrilateral, being given two opposite angles, the diagonals, and the angle between the diagonals. 54. A circle rolls inside another of double its diameter; find the locus of a fixed point in its circumference. 55. Two points, C, D, in the circumference of a given circle are on the same side of a given diameter; find a point P in the circumference at the other side of the given diameter, AB, such that PC, PD may cut AB at equal distances from the centre. 56. If the sides of any polygon be cut by a transversal, the product of one set of alternate segments is equal to the product of the remaining set.