IN^EST'IGJTION of PORISMS. 199 



PROP. VIL P OR ISM. Fig. 13. 



36. Three 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 able a given quadrilateral. Let A be a given point 

 in the line AD, and let ABLG be a quadrilateral, fimilar to the 

 given quadrilateral able, placed, fo that the angles of the tri- 

 angle ABC, fimilar to the given triangle abc, 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 CBlT will alfo be given in pofition and magnitude, 

 and the point L will be given. The line to be found muft 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 able, having its angles on the four lines LM, CF, BE and 

 AD, the angle at M being equal to the angle CLB, &c. 



Complete 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 to one another, are therefore given. Q. E. D. 



Cor. Hence alfo it appears, how a triangle given in fpecies 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 the lines. The folution of this problem is therefore taken for granted, in the 

 analyfis of the Porifm, though, for the fake of brevity, the conftruflion is omitted. 



