Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

APPLICATIONS

61

These points lie on a straight line. Let the student try to
prove this theorem by Plane Geometry.
The proof by Analytic Geometry is given immediately as a
direct application of the second of the general principles enunciated in the opening paragraph of the chapter.
Write down the equation of the line through two of these
points, - say, through the first and third. It is found to be:
3x-3y + 4 = 0.
The coordinates of the second point,
x =-0, y = 4,
are seen to satisfy this equation, and the proposition is proved.
EXERCISES
1. Prove the proposition for the general case (Fig. 2). The
points have been found to be:
(a - b c (. ab; +b ab + c).
3 3 2 2c
2. On plotting the three points obtained in the special case
discussed in the text it is observed that the line-segment determined by the extreme points is divided by the intermediate point in the ratio of 1: 2. Prove this analytically. Is it
true in general?
EXERCISES ON CHAPTER III
1. Prove that the three lines,
2x-3y —5=0, 3x+4y-16 =0, 4x-23y+7=0,
go through a point.
2. Prove that the three lines,
ax+ by=1, bx + ay=-1, x- y= O,
gp through a point.
3. Prove that the three points (4, 1), (- 1, - 9), and
(2, - 3) lie on a line.