194 On the ORIGIN and 



PROP. VI. PORISM. Fig. ii. 



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 interfect 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 a£ a given line, of which the feg- 

 ments a/3, (By, yd have given ratios to one another. 



Let A be a given point in the line AB, and fuppofe, that 

 AO is drawn from it, interfering 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 a/2, (By, yd of the line a5. Then, be- 

 caufe the lines CD, EF are given in pofition, and al fo 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 interfecls the lines 

 CD, EF and OL into the parts, BD, DF and FL, having the fame 

 ratios with the parts a/3, (By, yh. 



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 lemma, the 

 ratio of AB to EF. that is, of EG to EF is given. Through L 

 draw LN parallel to BG ; meeting GF produced in N. 



Then 



