ON THE INTERSECTION OF CONICS, AND ON PROJECTIONS. 1-3. Any two conics intersect in four points, real or imaginary. Ver- 4-7. If the four points of intersection be all real, or all imaginary, all the vertices are real. If two of the points of intersection be real, and two imaginary, one vertex only is real. If the four points of intersection be all real, all the common chords Any quadrilateral may be projected, in an infinite number of These projections may be effected in an infinite number of ways Any two intersecting conics may be projected into hyperbolas Any two conics may be projected into conics of any eccentricity, The anharmonic ratio of any four points on, or any four tangents Any system of points in involution projects into a system in involution, and the foci of one system project into the foci of A system of Conics, passing through four given points, cut any 3. If four points and one tangent be given, two conics can be ib. 16. The product of any two determinants is a determinant 17. Property of the co-ordinates of three points, forming a conjugate 18. Envelope of a side of an inscribed triangle whose other sides pass 19. Locus of a vertex of a circumscribed triangle, whose other vertices 9 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 a, 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, B, v, the trilinear co-ordinates of any point, then aa+bB+cy = 2A. Let P be the given point, and first suppose it to lie within F. 1 the triangle of reference (fig. 1). Join PA, PB, PC, and draw PD perpendicular to BC. Then PD-a, and aa = twice the area of the triangle PBC. Adding these equations, we get aa+bB+cy=2A. Next, suppose P to lie between AB, AC produced, and on the side of BC remote from A (fig. 2). Then a will be Fig. 2. B E . |