250 NOTE ON TRILTNEARS. 



Cor. 2. The case of a dividing point lying in a side produced is 

 included by making the ratio negative. 



2. The equations to a line. 

 Let (a, )8, y) be the coordinates of some arbitrary point in the line j 

 (a', j8 , y) current coordinates ; r the distance between these points ; 

 then we have 



I m n 



where I, m, n, are constants connected by the relation 



al + bm -\- en = 0, 

 This is obvious, because the numerators of the above ratios are the- 

 projections of r on lines perpendicular to the sides of the triangle. 

 Also since 



a a + fj j3^ -\- c y = 2 Area of triangle, 

 u a + b (3 + c y = same, 



it follows that 



a{a—a) + h{l3'—j3) + c{y—y) = 0, 

 and therefore 



al + bm + cw = 0. 



Hence also the equations to a line which passes through two point»- 



a — a /3' — ^ y — y 



a— a, /3-^i y— Vi * 



3: The tangent to a conic. 

 Let ^(a, ^, y") = G, be the general equation to the conic, ^ being 

 homogeneous and of the second order. The tangent to this at the 

 point (a, p, y) being tlie line through the points (a, ^, y) and (a + da, 

 ^ + dp, y + dy), its equations will be 



a — a, /8' — (i y' — y , 



da d/3 dy 



But, from the equation to the curve, 



''t . Ja + f^. d/3 + f . .!y = 0. 

 d a dp d y 



therefore the equation to the tangent becomes 



(cz'-a) f" + (i8-/5) ^ + (y'-y) ^J. = 0, 

 d a dp dy 



