17. If two circles touch externally, their common tangent is a mean proportional between their diameters.
18. If there be given three parallel lines, and two fixed points A, B; then if the lines of connexion of A and B to any variable point in one of the parallels intersect the other parallels in the points C and D, E and F, respectively, CF and DE pass each through a fixed point.
19. If a system of circles pass through two fixed points, any two secants passing through one of the points are cut proportionally by the circles.
20. Find a point O in the plane of a triangle ABC, such that the diameters of the three circles, about the triangles OAB, OBC, OCA, may be in the ratios of three given lines.
21. ABCD is a cyclic quadrilateral: the lines AB, AD, and the point C, are given in position; find the locus of the point which divides BD in a given ratio.
22. CA, CB are two tangents to a circle; BE is perpendicular to AD, the diameter through A; prove that CD bisects BE.
23. If three lines from the vertices of a triangle ABC to any interior point O meet the opposite sides in the points A′, B′, C′; prove
24. If three concurrent lines OA, OB, OC be cut by two transversals in the two systems of points A, B, C; A′, B′, C′, respectively: prove
