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

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

Let

$OA=\alpha ,OB=\beta ,OC=\gamma$;

then

$AB=\beta -\alpha$,

$AC=\gamma -\alpha$.

But $AC$ is a multiple of $AB$, 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/,