ANALYTIC QEOMi:n;\ 



the two given tixr.l points be P\**(x\*y\) and PjS 

 , and let P=(s, y) l>e any ntlui point on tin- lnu- 

 through P| and P,. Draw the ordinates 4f|P|, AT./V. ;unl 



/; 



M .!/. 



B 



Fio.40.^ 



IfP; also through Pj draw P^ 2 parallel to the z-axis, and 

 meeting MP in R and M^P 2 in R 2 . Thru tin- triangles 

 and P|/J 2 P 2 are similar ; 



JfP-Ar,P, OM-OM. 



i.e. _ * 



JfflnAn ./UlJll \J M. 9 ~~ \SJ.H.\ 



Substituting in this last equation the counlinatrs <>!' /', 

 2, and P, it becomes 



and since P= (x, y) is any point on the line through PI an 1 

 P,, therefore equation [9] is satisfied by the coordinates of 

 every point on this line. That equation [9] is not satisfied 

 by the coordinates of any point except such as are on the 

 line PiPj may be proved as was done in Art. 43. 



Equation [9] then fulfills both requirements of the defi- 

 nition in (1) of Art. 35, and is therefore the equation of 

 the straight line through the two points (x^ y\) and (z* y 2 ). 

 This equation will be frequently needed and will be referred 

 to as a standard form ; it should be committed to memory.* 



Throughout this book the more important formulas are printed in bold- 

 faced type ; they should be committed to memory by the learner. 



