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Additional info for A treatise on the analytical geometry of the point, line, circle, and conical sections (1885)

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Let P1 and P2 be two points with respective distances from BC equal to x¯1 and x¯2 , and let P be a point on P1 P2 such that P1 P : P1 P2 = λ. 3) which is the parametric representation of the line joining P1 = (x1 , y1 , z1 ) and P2 = (x2 , y2 , z2 ). 4) or, in the form of a determinant, x y z x1 y1 z1 = 0 . 5) This leads to the conclusion that in trilinear coordinates, the equation of a line is linear (and homogeneous). 4), we deduce that the equation is not only homogeneous in x, y, and z, but also in x1 , y1 , and z1 , and in x2 , y2 , and z2 .

At the creation of Carthage it may have influenced the actions of the astute queen Dido who was allowed to choose a piece of land, on condition that she could enclose it with an ox hide. To obtain the greatest possible area of land, she had the hide cut up into very thin strips that were then sown together and laid out in a half-circle, with a piece of shore as diameter [Vir]. 6 The following simple proof of the isoperimetric inequality for polygons, from which a proof for arbitrary curves follows by taking limits, was given by Bol (1943) [Bol2].

There is a simple relation between trilinear and barycentric coordinates: ¯ = a¯ X x, Y¯ = b¯ y , Z¯ = c¯ z. 7) then X : Y : Z = ax : by : cz . 8) In the coordinates X, Y , and Z, the line at infinity has equation X+Y +Z = 0. In these coordinates, a line has a homogeneous linear equation, and every homogeneous linear equation represents a line. 3 Let us determine the trilinear coordinates of a number of special points, and the equations of a number of lines. 8). The vertices A, B, and C are (1, 0, 0), (0, 1, 0), and (0, 0, 1), respectively, and the sides are given by x = 0, y = 0, and z = 0.

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