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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


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, , y, the trilinear co-ordinates of any point, then

ac + b3 + cy=24. Let P be the given point, and first suppose it to lie within

B and

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 bB = twice the area of PCA,

cy=twice the area of PAB. Adding these equations, we get

aa + b3 + 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.


negative, while B, y are positive. Hence, twice the area PBC will be represented by — ad, and we shall therefore have as before

aa + b3 + cy=2A. Thirdly, let P lie between AB, AC, produced backwards (fig. 3), so that B, y are negative while a is positive. Twice

Fig. 3.


the areas of PBC, PCA, PAB, are now represented by aa, - , - cy respectively, so that we still have

aa + b3 + cy=2A. In all cases, therefore,

aa + b3 + 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, ß and y for example, and this may be made homo

aa+bB+ oy geneous in a, b, y by multiplying each term by


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