The elements of coordinate geometry, by S. L. Loney, M.A.

CONCURRENCE OF STRAIGHT LINES.

55

79. To find the condition that three straight lines may
meet in a point.
Let their equations be
a~x + bly + c1 = O......... (1......... ),
a2x + b2y +c =2O.................(2),
and acx + bNy + c30.................(3).
By Art. 77 the coordinates of the point of intersection
of (1) and (2) are
b~c2 - bjzc1 c~a2 - c~a1
- and.......... (4).
aCb2 - a2b1 a,b2 - a2b(4
If the three straight lines meet in a point, the point of
intersection of (1) and (2) must lie on (3). Hence the
values (4) must satisfy (3), so that
61C2 - 69cC ca2 - C2A1
a. x --- + b. x - + C3 = 0)
alb2-a2b1 alb2 - a2b,
i. e. a. (blc2 - b2c1) + b3 (cla, - c2a3) + c3 (a,b, - a2b) = 0,
i. e. a, (b2c3 - b3c2) + b, (c2a3 - c,a2) -+- c (a2b3 - a,b2) 0... (5).
Aliter. If the three straight lines meet in a point let
it be (x,, y,), so that the values x, and y, satisfy the
equations (1), (2), and (3), and hence
a"x, + bly, + c1 0,
a2x1 + b2y1 ~ C2 0,
and acx, + b3y1 C3 0z
The condition that these three equations should hold
between the two quantities x, and y, is, as in Art. 12,
a1, bi, c1
a2, b2, C2 =-0
a3, 63, C3
which is the same as equation (5).
80. Another criterion as to whether the three straight
lines of the previous article meet in a point is the following.
If any three quantities p, q, and r can be found so
that
p (a1x ~ bly + c3) + q (a2x + b2y + c2) ~ r (a3x ~ b~y + c,) = 0
identically, then the three straight lines meet in a point.