90 Alt A l. ) 1 !< QEOMEin Y [Cii. V. 



By subtracting eq. (8) from eq ( - ) ami also CMJ. (4) from 

 eq. (3), the two equation s 



and 



are obtained. These give 



J: ... (5) 

 B 



.. (6) 



= -M l M^= 111', 

 and * 2 - * 8 



hence, from eq. (6), 

 Also, by construction, 



hence, triangle HP 2 P l is similar to triangle KP Z P V 

 and Z PP^T = 



-f Z ^TP 2 P 8 = 2 rt. zi ; 



i.e., P a lies on the straight line joining P 1 and P y lint. 

 since P 2 is any point on the locus of Ax + Ity + (7 = 0, hence 

 a// joints of this locus lie on the same straight line 1\T 

 which, therefore, constitutes the locus of Ax + By + C = 0. 

 Since this demonstration does not depend upon the angle 

 (o. therefore it applies whether the axes are oblique or rec- 

 tangular ; hence the theorem : every equation of the first 

 degree between two variables, when interpreted in Cartesian 

 coordinates^ represents a straight line.* 



' This conclusion may also be drawn thus : clear equation (6) of frac- 

 tions, transpose all the terms to the first member, and multiply by sin u ; 



