[ 507 ] 
A D. But thus the point E is in an hyperbola de- 
ferred to the afymptotes MN, MO, and paffin^ 
through A and D. ° 
THE determination of this locus for three lines is 
folved almoft explicitly by Apollonius in the three lalt 
propofitions of his third book of Conics. For if the 
three lines propofed were A B, AC, B C, and the 
point fought D, that the ratio of the redtangle under 
EDF (the line EF being drawn parallel 
to B C) fhould be in a given ratio to the fquare Fl §- 12 » 
of a line drawn from D to B C in a given I3, 14 * 
angle, the iquare of which line will be in a given 
ratio to the redtangle under BE, C F ; then if B H, 
Cl are drawn parallel to AC and A B refpedively, 
alfo BDL, CDK drawn through D, the fquare 
of B C will be to the redtangle under B K, CL as 
the redtangle under D F, DE, to that under C F, 
B E. 
Hence if the ratio of the redtangle under D F, 
D E to the fquare of a line drawn from D on B C 
in a given angle, is given; the fquare of this line 
being in a given ratio to the redtangle under C F, 
BE, the ratio of the redtangle under BK, CL to 
the fquare of B C will be given ; whence a conic 
fedtion palling through D will in all cafes be given. 
In the frit place let the point D be within the 
angle B A C. Tl hen if B C be bifedted by the line 
A M, this will be a diameter to the conic 
fedtion, which fhall touch BA, AC in the Fl S- I2 * 
points B, C, and B C will be ordinately applied to 
that diameter; the vertex of this diameter being N, 
the given ratio of the redtangle under BK, CL to 
2 the 
