SUPPLEMENTARY PROBLEMS. The product of any two determinants is a determinant Property of the co-ordinates of three points, forming a con- Envelope of a side of an inscribed triangle whose other sides 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 a, ẞ, 7. 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, y, the trilinear co-ordinates of any point, then Let P be the given point, and first suppose it to lie within 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. Fig. 1. Similarly bẞ = twice the area of PCA, cy twice the area of PAB. = 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. E negative, while B, y are positive. Hence, twice the area PBC will be represented by aa, and we shall therefore have as before aa+bB+cy = 2A. Thirdly, let P lie between AB, AC, produced backwards (fig. 3), so that B, y are negative while a is positive. Twice the areas of PBC, PCA, PAB, are now represented by aa, - bß, - cy respectively, so that we still have aa+bB+cy = 2A. In all cases, therefore, aa+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 homoaa+bB+cy geneous in a, ß, y by multiplying each term by 24 raised to a suitable power. Thus, the equation B2 + hy + k2 = 0 is equivalent to the homogeneous equation 4A3ß2 + 2Ahy (aa + bß + cy) + k2 (aa + bß+cy)2 = 0. The following examples may familiarize the reader with this system of co-ordinates. 1. Prove that the co-ordinates of the middle point of the 2. The co-ordinates of the centre of the circumscribed circle are R cos A, R cos B, R cos C, where 3. The co-ordinates of the centre of the inscribed circle are What are the co-ordinates of the centres of the escribed .circles? 4. The co-ordinates of the centre of gravity are 2A 2A 2A За 36 3c 5. Prove that a sin A + B sin B+ y sin C is equal to R where R is the radius of the circumscribing circle. 3. To find the distance between two given points, in terms of their trilinear co-ordinates. Let a,, B, Y, a,, B, Y, be the co-ordinates of two given points, r the distance between them. Then, 2 will be a rational integral function of α, — ɑ„ B1 - B2 Y1-Y2 of the second degree 1 * This, if not self-evident, may be proved as follows: Let P, Q be the two given points. Join PQ, and draw PM, QM' perpen |