194 On the ORIGIN and 
PRORIINVIEN. PORTIS M. PrGuetr. 
33. THREE ftraight lines being given in pofition, a fourth line, 
alfo given in pofition, may be found, fuch, that through 
any point whatever a ftraight line may be drawn, which 
will interfeét thefe four lines, and will be divided by them 
into three fegments, having given ratios to one another. 
Let AB, CD, EF be the three lines given in pofition, and OL 
the line to be found, and «da given line, of which the feg- 
ments «3, Sy, yd have given ratios to one another. 
Let A bea given point in the line AB, and fuppofe, that 
AO is drawn from it, interfeCting the lines CD, EF and OL 
in the points C, E and O, and divided at thefe points into the 
fegments AC, CE, EO, having the fame ratios to one another, 
with the given fegments «3, By, v2 of the line ad. Then, be- 
caufe the lines CD, EF are given in pofition, and alfo the point 
A, the line AE is given in pofition and magnitude, (§ 32.) and 
therefore alfo EO, which has a given ratio to AE; the point O 
is therefore given. 
AGAIN, let B be any point whatever in AB, and let BL be 
drawn, according to the hypothefis of the Porifm, fo as to be 
divided in the points D, F and L, where it interfects the lines 
CD, EF and OL into the parts, BD, DF and FL, having the fame 
ratios with the parts «8, By, yd. 
Let alfo BG be drawn equal and parallel to AE, and let EG 
be joined ; EG will therefore be parallel to AB, and will be 
given in pofition ; and if GF be drawn, it will make given 
angles with EG and EF, becaufe, by the preceding /emma, the 
ratio of AF to EF, that is, of EGto EF is given.. Through L ~ 
draw LN parallel to BG, meeting GF produced in N. 
THEN © 
