vee. 
Fe eae 
IEEE 
Bk ern 
INVESTIGATION of PORISMS. 199 
PE OP Nis oO RLS M. Fic. 13. 
36. TureeE ftraight lines being given in pofition, a fourth may 
be found, which will alfo be given in pofition, and will be 
fuch, that innumerable quadrilaterals, fimilar to the fame 
given quadrilateral, may be defcribed, having their angles 
placed, in the fame order, on the four ftraight lines given 
in pofition. 
Let AD, BE, CF be the three ftraight lines given in pofi- 
tion, and ad/c a given quadrilateral. Let A be a given point 
in the line AD, and let ABLC be a quadrilateral, fimilar to the 
given quadrilateral ab/c, placed, fo that the angles of the tri- 
angle ABC, fimilar to the given triangle a/c, may be, one of 
them, at the given point A, and the other two, on the lines BE 
and CF. The points B and C, and the triangle ABC, will there- 
fore be given, (Lemma 2. Cor.) and confequently the tri- 
angle CBL will alfo be given in pofition and magnitude, 
and the point L will be given. The line to be found mutt pafs 
through L; let it be LM; let M be any point in it whatfoever, 
and let MEDF be a quadrilateral fimilar to the given quadrila- 
teral abc, having its angles on the four lines LM, CF, BE and 
AD, the angle at M being equal to the angle CLB, fc. 
Compe ete the parallelogram AG, under CA, AD, and on DG 
defcribe the quadrilateral GDHN, fimilar and equal to the qua- 
drilateral 
is given, and the triangle CGK given in fpecies. The angle KGC is therefore given, 
and the angle KGF being alfo given, the angle CGF is given, and confequently the ratio 
of CG to CF. The ratios of the lines CG, CK and CF to one another, that is, of AD, 
BE and CF toone another, are therefore given. Q.E.D. 
Cor. Hence alfo it appears, how a triangle given in {pecies may be defcribed, having 
its angles on three ftraight lines given in pofition, and one of the angles at a given point 
in one of thelines. The folution of this problem is therefore taken for granted, in the 
analyfis of the Porifm, though, for the fake of brevity, the conftruGtion is omitted. 
