n* DEMONSTRATIONS of 



with AB, and are therefore given in pofition. But by Theor. 7. 



GB*+4- GC2 +-fGDH--f GE- = °~^GX^ = 

 HBM-^HD+^HDH-4hE* = SSSfifafc Now, 



a+b+c+J a+b+c+J, 



-— GX* = _JL( GX . +GY ,) _ 



7 — (GH*+HX 2 ), by Prop. 1. Therefore* 



HB»+~HC4-f HD"+4hE* = ^±^HX*. 



Again, by Theor. 7. FBM FO -f — FD* 4- -— FE* 



' ' 'a 'a 'a 



HB-+4-HC'+-^-HD'+4hE-+ ^±^HP ; therefore, 



FB>+ 4" FC«+ -f FD*+ -f FE ' = '-—^ (HX-+HF-), or, fince 



HX'+HF* = ^(FX'+FY 1 ), FB i +4fC'+4fD«+4- f E* = 



I+*£±-'(FV+FY»).. 



Cor, If from any point, as F, ftraight lines be drawn in given 

 angles to the lines which are given by pofition, and which interfect 

 in one point, two ftraight lines may be found which will be given 

 by pofition, fuch, that if perpendiculars from F be drawn to 

 them, the fum of the fquares of the lines drawn in given an- 

 gles, will be equal to the fpace to which the fum of the fquares 

 of the perpendiculars has a given ratio. 



This 



