200 On the RIO IN and 



drilateral ABLC ; join BH and LN, and it is evident, that the 

 three lines CG, BH and LN are all equal, and parallel to AD, and 

 are all given in pofition. Join alfo AL, DN, DM, MN and FG. 



Because the two quadrilaterals DEMF, DHNG are fimilar, 

 the angle FDM is equal to the angle GDN, and therefore the 

 angle GDF to the angle NDM. For the fame reafon alfo, 

 GD : DF : : ND : DM, and therefore the triangles GDF, NDM are 

 fimilar, and the angle FGD equal to the angle MND, and 

 FG : MN : : GD : DN, i'o that FG has a given ratio to MN. 



But becaufe the triangles ABC, DEF are fimilar, CG has a 

 given ratio to CF, (Lem. 2.) fo that the angle GCF being given, 

 the triangle CGF is given in fpecies, and FG has to GC a given 

 ratio \, now, FG was alfo (hewn to have to MN a given ratio \ 

 therefore MN has a given ratio to CG, that is, to LN. 



Again, fince the triangle CGF is given in fpecies, the angle 

 CGF is given, and CGD being alfo a given angle, the angle 

 FGD is given, and therefore MND, which is equal to it. But 

 the angle LND is given, therefore the angle LNM is given ; 

 and it was fhewn, that MN has a given ratio to NL, therefore 

 the angle MLN is given ; now, the point L, and the line LN, 

 are given in pofition ; therefore LM is alfo given in pofition, 

 which was to be found. 



The conftrudtion for finding LM is obvious. Take A and 

 D, two given points in one of the lines given in pofition, and 

 place the two triangles ABC, DEF fimilar to the given triangle 

 abc, fo that two of their equal angles may be at A and D, and 

 the other equal angles on the lines BE and CF, (Lem. 2. Cor.). 

 On BC and EF, defcribe the triangles BLC, FEM, fimilar to the 

 triangle cbl; if LM be drawn, it will be the line required. 



From the analyfis it alfo follows, that the quadrilaterals de- 

 fcribed with their angles on the four ftraight lines given in po- 

 fition, as fuppofed in the Porifm, will intercept between them 

 fegments of thefe lines, having given ratios to one another. 



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