50. AB is a diameter of a circle; AC, AD are two chords meeting the tangent at B in the points E, F respectively: prove that the points C, D, E, F are concyclic.
51. CD is a perpendicular from any point C in a semicircle on the diameter AB; EFG is a circle touching DB in E, CD in F, and the semicircle in G; prove—(1) that the points A, F, G are collinear; (2) that AC = AE.
52. Being given an obtuse-angled triangle, draw from the obtuse angle to the opposite side a line whose square shall be equal to the rectangle contained by the segments into which it divides the opposite side.
53. O is a point outside a circle whose centre is E; two perpendicular lines passing through O intercept chords AB, CD on the circle; then AB2 + CD2 + 4OE2 = 8R2.
54. The sum of the squares on the sides of a triangle is equal to twice the sum of the rectangles contained by each perpendicular and the portion of it comprised between the corresponding vertex and the orthocentre; also equal to 12R2 minus the sum of the squares of the distances of the orthocentre from the vertices.
55. If two circles touch in C, and if D be any point outside the circles at which their radii through C subtend equal angles, if DE, DF be tangent from D, DE.DF = DC2.
BOOK IV.
INSCRIPTION AND CIRCUMSCRIPTION OF TRIANGLES AND OF REGULAR POLYGONS IN AND ABOUT CIRCLES
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DEFINITIONS.
i. If two rectilineal figures be so related that the angular points of one lie on the sides of the other—1, the former is said to be inscribed in the latter; 2, the latter is said to be described about the former.