INVESTIGATION of PORISMS. i6y 



Again, becaufe the angle D F B is bifecled by F M, E M is 

 to M D as E F to F D, that is, in a given ratio ; and therefore, 

 fince E D is given, «E M, MD, are alfo given, and likewife the 

 point M. 



But becaufe the angle LFD ishalf of the angle EFD, and 

 the angle D FM half of the angle D F B, the two angles LFD, 

 D F M, are equal to the half of two right angles, that is, to a 

 right angle. The angle L F M being therefore a right angle, 

 and the points L and M being given, the point F is in the cir- 

 cumference of a circle defcribed on the diameter L M, and con- 

 fequently given in pofition. 



Now, the point F is alfo in the circumference of the given 

 circle ABC; it is therefore in the interferon of two given 

 circumferences, and therefore is found. 



Hence this conflruction : Divide E D in L, fo that EL may 

 be to L D in the given ratio of a, to (2 ; and produce E D alfo 

 to M, fo that E M may be to M D in the fame given ratio of a 

 to (3. Bifecl L M in N, and from the centre N, with the di- 

 itance N L, defcribe the femicircle L F M, and the point F, 

 in which it interfetits the circle A B C, is the point required, or 

 that from which F E and F D are to be drawn. 



The fynthetical demonftration follows fo readily from the 

 preceding analyfis, that it is not neceffary to be added. 



It muft however be remarked, that the conflrucYion fails 

 when the circle L F\M falls either wholly without, or wholly 

 within the circle ABC, fo that the circumferences do not in- 

 terfedl ; and in thefe cafes the folution is impoffible. It is plain 

 alfo, that in another cafe the conftruclion will fail, viz. when 

 it fo happens that the circumference L F M wholly coincides 

 with the circumference A B C. In this cafe, it is farther evi- 

 dent, that every point in the circumference ABC will anfwer 

 the conditions of the problem, which therefore admits of in- 

 numerable folurions, and may, as in the foregoing inflances, be 

 converted into a Porifm. 



13. We 



