The three straight lines AA', BB', CC', intersect in a point; and that The points of intersection of BC with B'C', CA with C'A', AB with A'B', lie in a straight line. 5. If two triangles circumscribe a conic, their angular points lie in another conic. 6. The equation of a conic circumscribing the triangle of reference, and having its semi-diameters parallel to the sides equal to r,,,,, respectively, is 7. A conic always touches the sides of a given triangle; prove that, if the sum of the squares on its axes be given, the locus of its centre is a circle, the centre of which is the point of intersection of the perpendiculars let fall from the angular points of the triangle on the opposite sides. 8. If be the angle between the asymptotes of the conic, represented by the general equation of the second degree, prove that 9. The two circular points at infinity may be represented by the equations, 99 LIBRARY OF THE OF MICH CHAPTER V. TRIANGULAR CO-ORDINATES. 1. WE shall now give a concise account of a system of co-ordinates which differs from that which has been the subject of the preceding chapters in assigning a slightly different interpretation to the co-ordinates. In the system which we are about to explain, the position of a point P is considered as determined by the ratios of the areas of the triangles PBC, PCA, PAB, to the triangle of reference ABC. If these quantities be denoted by the letters x, y, z, they will be connected by the identical relation x + y + z = 1. 2. In this method, as in that of trilinear co-ordinates, an equation of the first degree represents a straight line, and one of the second degree a conic. Again, since x: aa :: y: bß :: z cy, it follows that if the same straight line be represented in the two systems by the equations la + mB + ny = 0, l'x + m'y + n'z = 0; .. l : l'a :: m : m'b :: n : n'c. Hence we may pass from any relation among the coefficients in the trilinear system to that in the present one, by writing la, mb, nc, for l, m, n, respectively. Similarly, in conics, we may pass from any such formula to the corresponding one, by writing ua3, vb2, wc3, u'bc, v'ca, w'ab, for u, v, w, u', v', w'. we must write for U, b'U, and similarly for V and W, c'a2 V, a2b2 W. we must write for U', a2bcU', and similarly for V' and W', b'ca V', c2ab W'. Hence we obtain the following synopsis of formulæ : The straight lines drawn through the angular points of a triangle, bisecting the opposite sides, are represented by y-z=0, 2-x=0, x-y=0. The internal bisectors of the angles, by The perpendiculars, by y cot B-z cot C=0, z cot C-x cot A = 0, The distance between two points, by {a2 (y — y') (z' — z) + b2 ( z − z′) (x − x) + c2 (x − x') (y'− y)}3, or by - (b2+c2—a2) (x − x')2+(c2+a2—b2) (y—y')2+(a2+b2—c2) (z— {(b2+c2 2 The condition of parallelism of the straight lines 1, m, m' 1, n, n' or mn'-m'n+nl' — n'l + lm'-l'm=0. TRIANGULAR CO-ORDINATES. The condition of perpendicularity, 2ll'a2 + 2mm'b2 + 2nn'c2 - (mn' + m'n) (b2 + c3 — a3) — (nl′ + n'l) (c2 + a3 − b2) — (lm' + l'm) (a2 + b2 — c2) = 0, or '{(l — m) (l' — n') + (l — n) (l' — m')} a2 + {(n − 1) (n' — m') + (n − m) (n' — 1')} c2 = 0. 101 The perpendicular distance from the point (x, y, z) to the line lx+my+nz = 0, is (lx+my+ nz) 2A {(l — m) (1 — n) a2 + (m − n) (m − 1) b2 + (n − 1) (n − m) c2}‡ ° The line at infinity will be represented by x+y+z = 0. 3. Again, in conics we have the following formulæ : or if 1, 1, 1, 0 U+V+W+2U'+2 V'+2W'=0. A rectangular hyperbola, if ua2 + vb2 + wc3 — u' (b2 + c3 — a2) — v' (c2 + a2 — b3) — w' (a2 + b2 — c2) = 0, (u + u' — v'— w') a2 + (v + v' — w' — u') b2 |