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ratio to the redangle under the given line and E B 
or CF. 
Let the line given be H, and take M B and N C, 
that the redangle under M B C, and that under 
B C N be to that under B A and H in the given ra- 
tio of the rectangle under E D F to that under B E 
and H, B M and C N being equal. Then draw from 
M and N lines parallel to B A, C A, which fhall in- 
terfed E F in K and L, whereby, MK cutting 
C A in I, the redangle under MBC will be to that 
under B A and H as the redangle under BMC to 
that under M I and H, and alfo as the redangle un- 
der E K F to that under K I and H, that is, as the 
redangle under E D F to that under H and B E or 
M K, whence by adding the antecedents and confe- 
quents the redangle under K D L will be to the red- 
tangle under H and M I in the fame given ratio, 
which is alfo that of the redangle under BMC to 
the fame redangle under H and MI: the point D 
therefore, is in an hyperbola paffing through B and C 
having for afymptotes the lines M K and N L given in 
pofition, the redangle under KDL being equal to 
that under BMC, or that under M B N. 
If the two lines AB and A C are parallel, the 
locus may be known to be a parabola by the laft' 
propofition of the fourth book of Pappus. 
But if B C were parallel to one of the other, the 
locus will be an hyperbola, as the preceding, but 
affigned by a fhorter procefs. 
Suppofe the given lines to be A E, A F, 
and BC parallel to A F. And let the 
redangle under E D F be equal to that under D G, 
and the given line H, the line E G making given 
angles with A E, A F. Here take E I equal to H, 
i and 
