66. If A, B, C, D be four collinear points, find a point O in the same line with them such that OA.OD = OB.OC.
67. The four sides of a cyclic quadrilateral are given; construct it.
68. Being given two circles, find the locus of a point such that tangents from it to the circles may have a given ratio.
69. If four points A, B, C, D be collinear, find the locus of the point P at which AB and CD subtend equal angles.
70. If a circle touch internally two sides, CA, CB, of a triangle and its circumscribed circle, the distance from C to the point of contact on either side is a fourth proportional to the semiperimeter, and CA, CB.
71. State and prove the corresponding theorem for a circle touching the circumscribed circle externally and two sides produced.
72. Pascal’s Theorem.—If the opposite sides of an irregular hexagon ABCDEF inscribed in a circle be produced till they meet, the three points of intersection G, H, I are collinear.
Dem.—Join AD. Describe a circle about the triangle ADI, cutting the lines AF, CD produced, if necessary, in K and L. Join IK, KL, LI. Now, the angles KLG, FCG are each [III. xxi.] equal to the angle GAD. Hence they are equal. Therefore KL is parallel to CF. Similarly, LI is parallel to CH, and KI to FH; hence the triangles KLI, FCH are homothetic. Hence the lines joining corresponding vertices are concurrent. Therefore the points I, H, G are collinear.
73. If two sides of a triangle circumscribed to a given circle be given in position, but the third side variable, the circle described about the triangle touches a fixed circle.