41. In a circle, the point of intersection of the diagonals of any inscribed quadrilateral coincides with the point of intersection of the diagonals of the circumscribed quadrilateral, whose sides touch the circle at the angular points of the inscribed quadrilateral.
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 =
