We can shew in a similar manner that the equation - lx + mß + ny = 0, of DK is and that the equation of FH is la + mẞ - ny = 0. EXAMPLES. 1. The three bisectors of the angles of the triangle of reference have for equations, ß-y=0, y−a=0, and a— a-ẞ=0. 2. The three straight lines from the angular points of the triangle of reference to the middle points of the opposite sides have for equations bB-cy=0, cy-aa=0, and aa - bß=0. 3. If A'B'C' be the middle points of the sides of the triangle of reference, the equations of B'C', C'A', A'B' will be bẞ+cy-aa=0, cy+aa-bẞ=0, aa+bẞ-cy=0 respectively. 4. The equation of the line joining the centres of the inscribed and circumscribed circles of a triangle is a (cos B-cos C) +ẞ (cos C · cos A)+y (cos A cos B)=0. 5. Find the co-ordinates of the centres of the four circles which touch the sides of the triangle of reference. Find also the co-ordinates of the six middle points of the lines joining the four centres, and shew that the co-ordinates of these six points all satisfy the equation aßy+bya+caß=0. 6. If AO, BO, CO meet the sides of the triangle ABC in A', B', C'; and if B'C' meet BC in P, C'A' meet CA in Q, and A'B' meet AB in R; shew that P, Q, R are on a straight line. Shew also that BQ, CR, AA' meet in a point P'; CR, AP, BB' meet in a point Q'; and that AP, BQ, CC' meet in a point R'. 7. If through the middle points, A', B', C' of the sides of the triangle ABC lines A'P, B'Q, C'R be drawn perpendicular to the sides and equal to them; shew that AP, BQ, CR will meet in a point. 8. If p, q, r be the lengths of the perpendiculars from the angular points of the triangle of reference on any straight line; shew that the equation of the line will be apa+bqß+cry=0. 9. If there be two triangles such that the straight lines joining the corresponding angles meet in a point, then will the three intersections of corresponding sides lie on a straight line. [Let f, g, h be the co-ordinates of the point, referred to ABC one of the two triangles. Then the co-ordinates of the angular points of the other triangle A'BC' can be taken to be f', g, h; f, g', h and f, g, h' respectively. B'C' cuts BC where a=0 and = 0. Hence the three inter + β γ 260. The general equation of the second degree in trilinear co-ordinates, viz. ux2 + vß2 + wy2+2u'By + 2v'yx + 2w'aß = 0, is the equation of a conic section; for, if the equation be expressed in Cartesian co-ordinates the equation will be of the second degree. Also, since the equation contains five independent constants, these can be so determined that the curve represented by the equation will pass through five given points, and therefore will coincide with any given conic. 261. To find the equation of the tangent at any point of a conic. Let the equation of the conic be $ (a, B, y) = ua2 + vß2 + wy2 + 2u'By + 2v'ya + 2w'aß =0, and let a', B', '; a", B", y" be the co-ordinates of two points on it. The equation u (x — a′) (a — a’') + v (B — B') (B — B") + w (y +2u' (B − B') (y — y') + 2v′ ( y − y) (a — a′′) +2w' (a — a') (B — ß'') = $ (α, B, y), = = is really of the first degree in a, B, y, and therefore it is the equation of some straight line. The equation is satisfied by the values a a', BB', y=y, and also by the values a=a", B = B", y = y". Therefore it is the equation of the line joining the two points (a', B', y), (a", B", y"). Let now (a", B", y) move up to and ultimately coincide with (a', ', '), and we have the equation of the tangent at (a', B', y'), viz., uaa' + vßß' +wyy +u' (By' +yß') + v' (yx' + ay') + w' (aß′ + ßa') = 0. Using the notation of the Differential Calculus we may write the equation of the tangent at any point (a', B, Y') of the conic (a, B, y) = 0 in either of the forms 262. To find the equation of the polar of a given point. It may be shewn, exactly as in Art. 76, 100, or 118, that the equation of the polar of a point with respect to a conic is of the same form as the equation we have found for the tangent in Art. 261. 263. To find the condition that a given straight line may touch a conic. Let the equation of the given straight line be lx +mß +ny = 0 ̊... The equation of the tangent at (a', B', y') is a (ux' + w'ß' + v'y') +ß (w'a' + vß' + u'y') .(i). +y(v ́a' + u'ß'+wy') = 0...(ii). If (i) and (ii) represent the same straight line, we have ua'+w'B' + v'y' _ w'a' + vß' + u'y' _ v'a' + u'ß' + wy' = m n Putting each of these fractions equal to — λ, we have ux +w'B' + vỳ +λ =0, w'a' + vẞ' + u'y' + λm = 0, v'á' + ú'B' + wy' +λn = 0. Also, since (a', B', y) is on the line (l, m, n), Eliminating a, B', y', λ from these four equations we obtain the required condition or l2. (vw-u'2) + m2 (wu - v'2) + n2 (uv — w'2) +2mn (v'w' — uu') + 2nl (w'u' — vv') + 2lm (u'v′ — ww') = 0, or Ul2 + Vm2 + Wn2 + 2U'mn+2 V'nl + 2 W'lm = 0... (v), where U, V, W, U, V', W' are the minors of u, v, w, u', v', w' in the determinant 264. To find the co-ordinates of the centre of a conic. Since the polar of the centre of a conic is altogether at an infinite distance, its equation is But [Art. 262], the equation of the polar of the centre will be where do аф do Bo, Yo, are the co-ordinates of the centre. 265. To find the condition that the curve represented by the general equation of the second degree may be a parabola. The co-ordinates of the centre of the curve are given by the equations ux+w'ß。+v ́¥ _ w'a + vß + u ́Ÿ‚ _v'a+u'ß ̧+wy = C Put each of these equal to λ, and we have ux + w ́B + v'Y +λa = 0, Also since the centre of a parabola is at infinity, we The elimination of α, B., Y。, λ gives for the required condition We see from the above that the parabola touches the line at infinity. [Art. 263.] 266. To find the condition that the conic represented by the general equation of the second degree may be two straight lines. The required condition may be found as in Art. 37. The condition is 267. To find the asymptotes of a conic. The equations of the curve and of its asymptotes only differ by a constant. Hence if the equation of the curve be ux2+ vß2 + wy3 + 2u' By + 2v'ya + 2w'aß = 0, the equation of the asymptotes will be ua2 + vß2 + wy2 + Qu'ẞy + 2v'ya +2w'aß +λ(ax+bB+cy)2 = 0........(i). The value of λ is to be determined from the condition. |