

ANA I.) //' <- MM/ A;//;)' 



.(',.. IV. 



let P = (./-.//) IKJ any other point on the lino through /', 

 an.l i 



Draw tin- nrdinates M^P^ MP* and JW 3 P. 2 . and through 

 P, dniw P^N parallel to the .r-axis, meeting MT in A' and 

 in /*,. 



The triangles P^RP and P^R^P^ are similar, hem 

 MP - MP OM- 



' 



Suhstituting for JlfP, OAf, 

 this equation becoin- 



^ etc., their valu->, 



5-2 



hich reduces to 



a; 



This is the required equation of the straight line through 

 1\ and P y because it fulfills both the requirements of the 

 definition [cf. Art. 35 (1)]; i.e., it is satisfied \>y the coordi- 

 nates of any (i.e., of every) point of this line, because a:,y are 

 the < M'ird i nates of any such point; and it is not satisfied l.\ 

 the co<rdinates of any point which is not on this line, because 

 the corresponding constructions tW sueh a point would not 

 give similar triangles, and hence the proiM)rtions which led 

 .is equation would not l>c true. 



That equation (1) is not satisfied by the coordinates of 



