15. One of the vertices of a triangle of given form remains fixed; the locus of another is a right line or circle; find the locus of the third. 16. Find the area of a triangle—(1) in terms of its medians; (2) in terms of its perpendiculars. 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 25. The line joining the middle points of the diagonals of a quadrilateral circumscribed to a circle— divides each pair of opposite sides into inversely proportional segments; is divided by each pair of opposite lines into segments which, measured from the centre, are proportional to the sides; is divided by both pairs of opposite sides into segments which, measured from either diagonal, have the same ratio to each other. 26. If CD, CD′ be the internal and external bisectors of the angle C of the triangle ACB, the three rectangles AD.DB, AC.CB, AD.BD′ are proportional to the squares of AD, AC, AD′; and are—(1) in arithmetical progression if the difference of the base angles be equal to a right angle; (2) in geometrical progression if one base angle be right; (3) in harmonical progression if the sum of the base angles be equal to a right angle. 27. If a variable circle touch two fixed circles, the chord of contact passes through a fixed point on the line connecting the centres of the fixed circles.