on la + mß + ny. formula of Art. 34, &c., the result is Write the equation at full length and apply the when, if we write x' cos a + y' sin a − p = a', la' + mB' + ny' √ \l2 + m2 + n2 − 2mn cos A – 2nl cos B – 2lm cos C)* Ex. 1. To find the equation of a perpendicular to y through its extremity. The equation is of the form la+ny = 0. And the condition of this article gives n = l cos B, as in Ex. 6, p. 54. Ex. 2. To find the equation of a perpendicular toy through its middle point. The middle point being the intersection of y with a sin A - ẞ sin B, the equation of any line through it is of the form a sin A - ẞ sin B + ny = 0, and the condition of this article gives n = sin (A — B).. -- Ex. 3. The three perpendiculars at middle points of sides meet in a point. For eliminating a, ẞ, y in turn between a sin A – ẞ sin B + y sin (A − B) = 0, ẞ sin B – y sin C+ a sin (B − C') = 0, we get for the lines joining to the three vertices the intersection of two perpen α diculars COS A = = β γ ; and the symmetry of the equations proves that the third perpendicular passes through the same point. The equations of the perpendiculars vanish when multiplied by sin2C, sin2A, sin2B, and added together. Ex. 4. Find, by Art. 25, expressions for the sine, cosine, and tangent of the angle between la+mẞ + ny, l'a + m'ß + n'y. Ex. 5. Prove that a cos 4 + ẞ cos B + y cos C is perpendicular to -- a sin A cos A sin (B − C) + ẞ sin B cos B sin (CA) + y sin C cos C sin (A — B). Ex. 6. Find the equation of a line through the point a'ß'y' perpendicular to the line 7. Ans. a B'y' cos A) — ẞ (a' + y' cos B) + y (B' cos B a' cos 4). 62. We have seen that we can express the equation of any right line in the form la + mẞ+ny = 0, and so solve any problem by a set of equations expressed in terms of a, B, y, without any direct mention of x and y. This suggests a new way of looking at the principle laid down in Art. 60. Instead of regarding a as a mere abbreviation for the quantity x cosa + y sin a-p, we may look upon it as simply denoting the length of the perpendicular from a point on the line a. We may imagine a system of trilinear coordinates in which the position of a point is defined by its distances from three fixed lines, and in which the position of any right line is defined by a homogeneous equation between these distances, of the form la + mB + ny = 0. The advantage of trilinear coordinates is, that whereas in Cartesian (or x and y) coordinates the utmost simplification we can introduce is by choosing two of the most remarkable lines in the figure for axes of coordinates, we can in trilinear coordinates obtain still more simple expressions by choosing three of the most remarkable lines for the lines of reference a, B, y. The reader will compare the brevity of the expressious in Art. 54 with those corresponding in Chap. II. 63. The perpendiculars from any point 0 on a, B, y are connected by the relation aa+bB+cy = M, where a, b, c, are the sides, and M double the area, of the triangle of reference. For evidently aa, bß, cy are respectively double the areas of the triangles OBC, OCA, OAB. The reader may suppose that this is only true if the point O be taken within the triangle; but he is to remember that if the point O were on the other side of any of the lines of reference (a), we must give a negative sign to that perpendicular, and the quantity aa+bB+cy would then be double OCA + OAB- OBC, that is, still double the area of the triangle. Since sin A is proportional to a, it is plain that a sin A + B sin B+ y sin C is also constant, a theorem which may otherwise be proved by writing a, B, y at full length, as in Art. 60, multiplying by sin(8-y), sin (y- a), sin (a-B), respectively, and adding, when the coefficients of x and y vanish, and the sum is therefore constant. = The theorem of this article enables us always to use homogeneous equations in a, B, y, for if we are given such an equation as a 3, we can throw it into the homogeneous form = 64. To express in trilinear coordinates the equation of the parallel to a given line la + mß +ny. It In Cartesian coordinates two lines Ax + By + C, Ax + By + C', are parallel if their equations differ only by a constant. follows then that la + mẞ + ny+k (a sin A+B sin B+ y sin C) = 0 denotes a line parallel to la+ mẞ+ny, since the two equations differ only by a quantity which has been just proved to be constant. In the same case Ax + By + C+ (Ax+ By + C') denotes a line also parallel to the two given lines and half-way between them; hence if two equations P=0, P'=0 are so connected that P- P' constant, then P+ P' denotes a parallel to P and P' half-way between them. = Ex. 1. To find the equation of a parallel to the base of a triangle drawn through the vertex. Ans. a sin A+ ẞ sin B = 0. For this, obviously, is a line through aß; and writing the equation in the form y sin C (a sin A + ẞ sin B + γ sin C') = 0, it appears that it differs only by a constant from y = 0. We see, also, that the parallel a sin A+ẞ sin B, and the bisector of the base a sin A - ẞ sin B, form a harmonic pencil with a, ß, (Art. 57). Ex. 2. The line joining the middle points of sides of a triangle is parallel to the base. Its equation (see Ex. 2, p. 58) is a sin A + ẞ sin B- y sin C = 0, or 2y sin C = a sin A + ẞ sin B + γ sin C. Ex. 3. The line aa - bẞ + cy - do (see Ex. 5, Art. 54) passes through the middle point of the line joining ay, ẞd. For (aa+cy) + (bẞ + dô) is constant, being twice the area of the quadrilateral; hence aa +cy, bß + dò are parallel, and (aa +cy) − (bß + dô) is also parallel and half-way between them. It therefore bisects the line joining (ay), which is a point on the first line, to (68) which is a point on the second. - 65. To write in the form la + mB+ny = 0 the equation of the line joining two given points x'y', x"y". Let a', as before, denote the quantity x' cos a + y' sin a — p. Then the condition that the coordinates x'y' shall satisfy the equation la+mẞ+ ny=0 may be written. in the given form, we obtain for the equation of the line joining the two points a (B'y" — y'ß') + B (y'a" – y'a') + y (a'ß" — a′′B') = 0. It is to be observed that the equations in trilinear coordinates being homogeneous, we are not concerned with the actual lengths of the perpendiculars from any point on the lines of reference, but only with their mutual ratios. Thus the preceding equation is not altered if we write pa', pß', py', for a', B', y'. Accordingly, if a point be given as the intersection of the lines α B ช we may take l, m, n as the trilinear coordinates 7 = m = n of that point. For let p be the common value of these fractions, and the actual lengths of the perpendiculars on a, B, y are lp, mp, np, where p is given by the equation alp + bmp + cnp = M, but, as has been just proved, we do not need to determine p. Thus, in applying the equation of this article, we may take for the coordinates of intersection of bisectors of sides, sin B sin C, sin C sin A, sin A sin B; of intersection of perpendiculars, cos B cos C, cos C cos A, cos A cos B; of centre of inscribed circle 1, 1, 1; of centre of circumscribing circle cos A, cos B, cos C, &c. Ex. 1. Find the equation of the line joining intersections of perpendiculars, and of bisectors of sides (see Art. 61, Ex. 5). Ans. a sin A cos A sin (B - C) +ẞ sin B cos B sin (C−A) + y sin C cos C sin (A-B) = 0. Ex. 2, Find equation of line joining centres of inscribed and circumscribing circles. Ans. a (cos B-cos C) + B (cos C - cos A) + y (cos A cos B) = 0. 7+ m 66. It is proved, as in Art. 7, that the length of the perpendicular on a from the point which divides in the ratio : m the line joining two points whose perpendiculars are a', a" is la' + ma" Consequently the coordinates of the point dividing in the ratio :m the line joining a'B'y', a"B"y" are la' + ma", iB' + mß", ly' + my". It is otherwise evident that this point lies on the line joining the given points, for if a'ß'y', a"ß"y" both satisfy the equation of a line Aa + BB+ Cy = 0, so will also la+ma", &c. It follows hence, without difficulty, that la - ma", &c., is the fourth harmonic to la' + ma", a', a"; that the anharmonic ratio of a-ka", a' — la", a' — ma", a' —na" is (n − 1) (m −k) and also that, given two systems of points on (n − m) (l−k) two right lines a' - ka", a' - la", &c., a"" — ka"""', a" — la"", la"", &c.; these systems are homographic, the anharmonic ratio of any four points on one line being equal to that of the four corresponding points on the other. Ex. The intersection of perpendiculars, of bisectors of sides, and the centre of circumscribing circle lie on a right line. For the coordinates of these points are cos B cos C, &c., sin B sin C, &c., and cos A, &c. But the last set of coordinates may be written sin B sin C cos B cos C, &c. The point whose coordinates are cos (BC), cos (C– A), cos (A – B) evidently lies on the same right line and is a fourth harmonic to the three preceding. It will be found hereafter that this is the centre of the circle through the middle points of the sides. 67. To examine what line is denoted by the equation a sin A+ẞ sin B+ y sin C = 0. This equation is included in the general form of an equation of a right line, but we have seen (Art. 63) that the left-hand member is constant, and never = 0. Let us return, however, to the general equation of the right line Ax+ By + C=0. We saw that the intercepts cut off on the axes are C с A' - Bi consequently, the smaller A and B become the greater will be the intercepts on the axes, and therefore the more remote the line represented. Let A and B be both = 0, then the intercepts. become infinite, and the line is altogether situated at an infinite distance from the origin. Now it was proved (Art. 63) that the equation under consideration is equivalent to Ox + Oy + C = 0, and though it cannot be satisfied by any finite values of the coordinates, it may by infinite values, since the product of nothing by infinity may be finite. It appears then that a sinA+ß sinß + y sin C denotes a right line situated altogether at an infinite distance from the origin; and that the equation of an infinitely distant right line, in Cartesian coordinates, is 0.x +0.y + C = 0. We shall, for shortness, commonly cite the latter equation in the less accurate form C=0. 68. We saw (Art. 64) that a line parallel to the line a = 0 has an equation of the form a + C=0. Now the last Article shows that this is only an additional illustration of the principle of Art. 40. For a parallel to a may be considered as intersecting it at an infinite distance, but (Art. 40) an equation of the form a+C=0 represents a line through the intersection of the lines. a=0, C=0, or (Art. 67) through the intersection of the line a with the line at infinity. 69. We have to add that Cartesian coordinates are only a particular case of trilinear. There appears, at first sight, to be an essential difference between them, since trilinear equations are always homogeneous, while we are accustomed to speak of Cartesian equations as containing an absolute term, terms of the first degree, terms of the second degree, &c. A little reflection, however, will show that this difference is only apparent, and |