Page:Linear Algebra (1882) Tevfik.djvu/25

 same values of $$a, b, c$$ which render $$a\alpha + b\beta + c\gamma = 0$$ will also render $$a + b + c = 0$$.

Let $O A = \alpha, O B = \beta, O C = \gamma$;

then

$A B = \beta - \alpha$,

$A C = \gamma - \alpha$.

But $$A C$$ is a multiple of $$A B$$, or

$\gamma - \alpha = x(\beta - \alpha) = x\beta - x\alpha$;

$$\therefore$$

$x\alpha - \alpha - x\beta + \gamma = 0$,

or

$(x - 1)\alpha - x\beta + \gamma = 0$;

and as in this equation the coefficients of a, p, y are x — 1, — a:, +1 which correspond to a, 6, c in the first equation, and as (x — 1) — aj+1=:0, then a-h6 + c=:0. 22. Conversely, if «, P, y are coinitial, coplanar lines, and if both a«-H&p + cY=0, and a-H6H-c=0, then do «, P, y terminate in a straight line. For by supposition, a-f-fc-hc = 0, therefore aY-*-6Y-»-CY=0, and by subtraction a(Y-«)-H6(Y — P) = or (^_a) + ^(Y_p) = 0. This shows that y — « is a multiple of y — P and therefore it is in the same straight line with it; «, p, y terminate in that straight line. 23. Examples. Ex. 1. In a plane triangle are given one angle, an adjacent side, and the sum of the lengths of the other sides, to determine the triangle. Let ABD be the given angle, AB = b ,, ,, ,, side, S MM sum of the lengths of the other two sides. 8 If in designating by a and p two unit lines, we represent by a; « the unknown side adjacent to the angle B, and by t/p the opposite side to this angle, we shall have j/P=6-hx<x and S = a? + 2/,