INVESTIGATION of PORISMS. 195 



Then becaufe the triangles BFG, LFN are fimilar, GF is to 

 FN as BF to FL, that is, in a given ratio ; and therefore, fince 

 FG alfo makes given angles with the two ftraight lines EG and 

 EF, given in pofition, the point N is in a ftraight line, given in 

 pofition, and paffing through E, viz. EN. 



Now, fince BF is to FL as BG to LN, and alfo as AE or BG 

 to EO, LN and EO are equal, and being alfo parallel, OL is 

 parallel to EN, that is, to a line given in pofition ; and the 

 point O, in OL, is given, therefore OL is given in pofition ; 

 which was to be found. 



Conjiruffiion. From any two given points, A and B', in the 

 line AB, draw AE and B'F interfeding CD and EF in C, E, D' 

 and F, fo that AC may be to CE, and B'D' to D'F' in the fame 

 given ratio of a/3 to (By, (§32.) Produce alfo AE to O, and 

 B'F to L', fo that AE may be to EO, and B'F to FL' in the 

 fame given ratio of ay to yh. If OL' be joined, it will be the 

 line required. 



For let B be any point whatfoever in AB, and as AB' to AB, 

 fo let OL' be to OL, and let BL be drawn, cutting CD', and 

 EF' in D and F, the line BL is divided in thefe points, fimilarly 

 to the given line ah. For fince the two lines AO and B'L' are 

 divided fimilarly by the three lines AB', CD' and OL', and fince 

 two of thefe laft, AB' and OL', are alfo divided fimilarly to one 

 another by the three lines AO, B'L' and BL, BL will be divided 

 in D, in the fame ratio wherein B'L' is divided in D', or AO in C, 

 (Lem. 1. Conv.). In the fame way, BL is divided in F, in the 

 fame ratio wherein AO is divided in E ; BL is therefore fimilarly 

 divided to AO, or to a£, which was to be demonftrated. 



34. Hence it is plain, " If two fimilarly divided lines, as AO 

 and BL, be drawn any how, and if ftraight lines AB, CD, EF, 

 OL, be drawn through the points of divifion of thefe lines, innu- 

 merable lines may be placed between the lines AB, CD, EF and 

 OL, which will be divided by them, fimilarly to the lines AO, and 

 BL." For, by what is here demonftrated, every line which cuts 



B b 2 any 



