88 ANALYTIC GEOMETRY II. 



through P l and P 8 draw lines parallel to 

 "A". UK M -ting 3f 8 P 8 and J/j/^ in J2 and # respectively. 



To find OM Z = * 8 and M^P Z = y 8 in terms <f r r .r.,, ?/ r y 2 , 

 ;/,. and tn s . 



The triani^lrs P^P^ and P Z QP^ are similar; 



P*I\, J\ I o / | 



and 



[In Firr. 14(5), x r y v y r and // 3 are negative.] 



a: a:, 7/a ?/ t m, 

 therefore _1 = ^ ^1 = _J; 



^2 - *s y - ^8 w s 

 vrhenoe 



-- 



The above reasoning applies equally well whatever the 

 Talue of (a (the angle made by the coordinate axes), hence 

 formulas [6] hold whether the axes be rectangular or oblique. 



Formulas [6] were obtained on the implied hypotlx -i- 

 that P 8 lies between P l and P 2 ; i.e., that P 8 is an internal 

 point of division. If P 8 is taken in the line P^P* j>r<lu< < <!. 

 and not between P l and P 2 , it still forms, with P l and P 2 , 

 two segments P^P Z and P 8 P 2 , and P 8 may be so tak< n that. 

 numerically, tin- ratio of PjP 8 : P 8 P a may have any n-al 

 value whatever ; l)iit the sign of this ratio is negative when 

 P 8 is not between P l and P 2 , for, in that case, the segments 

 PjP 8 and P 8 P a have opposite directions. Hence, to find 

 tin- coordinates of that point which divides a line externally 

 into segments whose numerical ratio is m 1 : m v it is only 



