Any system of points in involution projects into a system in involution, and the foci of one system project into the foci If five points be given, one conic only can be drawn If one point and four tangents be given, two conics can be SUPPLEMENTARY PROBLEMS. The product of any two determinants is a determinant Property of the co-ordinates of three points, forming a cón- Envelope of a side of an inscribed triangle whose other sides pass each through a fixed point, Locus of a vertex of a circumscribed triangle, whose other ver- TRILINEAR CO-ORDINATES. CHAPTER I. TRILINEAR CO-ORDINATES. EQUATION OF A STRAIGHT LINE. 1. IN the system of co-ordinates ordinarily used, the position of a point in a plane is determined by means of its distances from two given straight lines. In the system of which we are about to treat, the position of a point in a plane will be determined by the ratios of its distances from three given straight lines in that plane, these straight lines not passing through the same point. The triangle formed by these three straight lines is called the triangle of reference, its sides, lines of reference, and the distances of a point from its three sides will be called the trilinear co-ordinates of that point. We shall usually denote the angular points of the triangle of reference by the letters A, B, C, the lengths of the sides respectively opposite to them by a, b, c, and the distances of any point from BC, CA, AB respectively by the letters α, B, Y. When two points lie on opposite sides of a line of reference, the distance of one of these points from that line may be considered as positive, and that of the other as negative. We shall consider a, the distance of a point from the line BC, as positive if the point lie on the same side of that line as the point A does, negative if on the other side; and similarly for B and y. It thus appears that the trilinear co-ordinates of any point within the triangle of reference are all positive; while no point has all its co-ordinates negative. 2. Between the trilinear co-ordinates of any point an important relation exists, which we proceed to investigate. If A denote the area of the triangle of reference, a, ß, y, the trilinear co-ordinates of any point, then ax+bB+cy = 2A. Let P be the given point, and first suppose it to lie within F. LIBRARY OF THE 1 the triangle of reference (fig. 1). Join PA, PB, PC, and Then PD = α, and ax = twice draw PD perpendicular to BC. the area of the triangle PBC. Next, suppose P to lie between AB, AC produced, and on the side of BC remote from A (fig. 2). Fig. 2. Then a will be B negative, while B, y are positive. Hence, twice the area PBC will be represented by az, and we shall therefore have as before aa+bB+cy = 2A. Thirdly, let P lie between AB, AC, produced backwards (fig. 3), so that ß, y are negative while a is positive. Twice the areas of PBC, PCA, PAB, are now represented by uz, bß, -cy respectively, so that we still have aa+bB+cy = 2A. In all cases, therefore, ax+bB+cy=2A. The importance of the above proposition arises from its enabling us to express any equation in a form homogeneous with respect to the trilinear co-ordinates of any point to which it relates. Any locus may be represented, as in the ordinary system, by means of a relation between two coordinates, B and y for example, and this may be made homoax + bB + cy geneous in a, B, y by multiplying each term by 2A |