MOORE. — MINIMUM GEOMETRY. 



203 



that is the distance between two points is the same as the distance 

 between the points into which they project from F. 



With this system of coordinates the equation y = kx, represents a 

 line through 0. For let 0Q be a line passing through 0, Figure 1, 



Figure 1. 



and let {x\, i/i) be the coordinates of R and (x, y), the coordinates 

 of Q. Then from the properties of distance it is seen at once that, 

 0Q = OS, OR = 0T. From the definition of measure on OF and the 

 comparison of two distances on any line we see that, 



0L : 0M = 0Q : OR 

 Therefore 



0L 

 0M 



Ob y x ?/, 



j or - = - > or y = - x. 



?/i xi xi 



0T 



In the system of measure established on / we will take the coordinates 

 of the point in which the line y = kx cuts / as k. Then the coefficient 

 k will represent the angle which 0Q makes with OP. 



To find the equation of a line joining two points, we will take as 

 the defining property of the line the fact that the area of the triangle 

 formed by any three of its points is zero. Let Qi(.i*i, yi), Q2OT2, 2/2) be 

 two fixed points and Q (x, y) be any point on the line joining them, then 



QQ1Q2 = OQQ1+ 0QQ 2 - OQ2Q1. 

 Using the formula for area this becomes 



x xj \x x 2 J \x 2 xj 



or (x - Xi) {yi - y 2 ) = (y - iji) Oi - .t 2 ). 



That is the equation of the straight line is of the first degree. 



