converted into a Porifin. 
INVESTIGATION o PORISMS. 167 
AGAIN, becaufe the angle DF B is bifeed by FM, EM is 
to M Das EF to FD, that is, in a given ratio; and therefore, 
fince E D is given, EM, MD, are alfo given, and likewife the 
point M. 
Bur becaufe the angle L F D is half of the angle E FD, and 
‘the angle DFM half of the angle D F B, the two angles L F D, 
DF M, are equal to the half of two right angles, that is, to a 
right angle. The angle I. FM 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, ped con- 
fequently given in pofition. 
Now, the point F is alfo in the circumference of the given 
circle ABC; it is therefore in the interfeGion of two given 
circumferences, and therefore is found. 
Hence this conftruction: Divide E D in L, fo that EL may 
be to LD in the given ratio of « to @; and produce ED alfo 
to M, fo that EM may be to M D in the fame given ratio of « 
to 6. Bifect LM in N, and from the centre N, with the di- 
ftance NL, defcribe the femicircle LF M, and the point F, 
in which it interfects the circle A BG, is the point required, or 
that from which F E and F D are to be drawn. 
Tue fynthetical demonftration follows fo readily from the 
preceding analyfis, that it is not neceflary to be added. 
_ Iv mutt however be remarked, that the conftruGtion fails 
when the circle LF.M falls either wholly without, or wholly 
within the circle ABC, fo that the circumferences do not in- 
terfecét ; and in thefe cafes the folution is impoflible. It is plain 
alfo, that in another cafe the conftrution will fail, viz. when 
it fo happens that the circumference L FM wholly coincides 
_ with the circumference ABC. 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 folutions, and may, as in the foregoing inftances, be 
13. WE 
