investigate the relation between the co-ordinates (a, B, y) of the point P. Fig. 6. B The property of the straight line, which we shall make the basis of our investigation, is, that it is the locus of a point which moves in such a manner, that the sum of the areas of the triangles PAQ, PAR is constant. Let AQ=q, AR=r, then the areas of the triangles PAQ, PAR will be respectively represented by qß, try, and the area of QAR by A. Hence = qr bc 2qr qB+ry= A qr bc -2 (aa+bB+cy). bc This is the equation of the straight line QR, and, since it involves the two arbitrary quantities q, r, it is in the most general form of the equation of the first degree between two variables. Putting the equation may be written la + mB + ny = 0. 8. We shall next establish the converse proposition, that every equation of the first degree represents a straight line. be the general equation of the first degree, and let f, g, h be the co-ordinates of any fixed point D on the locus of the equation, a, ẞ, y those of any point P. Draw DE, PM perpendicular to AC, DF, PN perpendicular to AB. Also draw Dm, Dn, perpendicular respectively to PM, PN. Then Pmẞ-g, Pn-y-h. Also, since f, g, h is a point on the locus, Hence, the ratio of Pm to Pn is constant, whatever point on the locus P may represent. This can only be true when. that locus is a straight line. 9. To find the co-ordinates of the point of intersection of two given straight lines. Let the equations of the two straight lines be give the values of a, B, y, at the point of intersection. 10. To find the equation of the straight line, passing through two given points. Let f, g, h; f, g, h, be the co-ordinates of the two given points, and suppose the equation of the required straight line to be = M N gh' - g'h hf' - h'ƒ ̄ ̄ƒg' -ƒ'g' giving, as the equation of the required line, (gh' — g'h) a + (hf' — 'h' ƒ) B+ ( ƒg′ − ƒ'g) y = 0. 11. To find the general equation of a straight line, passing through the point of intersection of two given straight lines. If the equations of the straight lines be la+mß + ny = 0, l'a + m2 B + n'y = 0, every straight line, passing through their point of intersection, may be represented by an equation of the form la + mẞ+ny = k (l'a + m'ẞ + n'y), where k is an arbitrary constant. For this equation is satisfied when the equations of the given straight lines are both satisfied, and, being of the first degree, it represents a straight line. It is therefore the equation of a straight line passing through their point of intersection. 12. To find the condition that three points may lie in the same straight line. Let a1, B11; α2, B2, V2 ; αg, B3, Y; be the co-ordinates of the three given points, then, if these points lie in the same straight line, suppose the equation of that line to be λα + β + γ = 0. Then λ, μ, v must satisfy the following equations: λα + μβ, + vy, = 0, λα + β + 2 = 0, λας + μβ + γ = 0, whence, eliminating λ, μ, v, by cross multiplication, the required condition. = 13. To find the condition that three straight lines may intersect in a point. Let the equations of the straight lines be 11a+m1ß + n1y = 0, la+m2ß + n2y=0, l ̧a+m ̧ß+n ̧y=0. If these three straight lines intersect in a point, the above three equations must be satisfied by the same values of a, B, y. This gives, eliminating a, B, y by cross-multiplication, 1 ̧m ̧ ̧− 1 ̧m ̧n ̧ +1 ̧m ̧n ̧ − l¿m ̧n ̧ + 1 ̧m ̧n— 1 ̧mn ̧ = 0, the required condition. The identity of form between the conditions that three straight lines should intersect in a point, and that three points should lie in a straight line, is worthy of notice. Its full geometrical meaning will be seen hereafter. We shall sometimes, in future investigations, speak of the straight line represented by the equation la + mẞ+ny = 0, as the straight line (1, m, n). Adopting this phraseology, it will be seen that the condition that the three points (1, m1, n1) (l2, m, n) (13, m ̧, n.) should lie in the same straight line, is the same as the condition that the three straight lines (11, m1, n1) (12, m, n2) (l, m, n,) should intersect in a point. 14. To find the condition that two straight lines may be parallel to one another. Let the equations of the two straight lines be Let (f, g, h) (a, B, y) be the co-ordinates of any two points in (1), (f', g', h') (a', B', y') be the co-ordinates of any two points in (2). |