The First Six Books of the Elements of Euclid

by Euclid; John Casey

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.