1806 On the ORFGIN and 
of the fquares of the perpendiculars drawn from it to AC, and 
CB, fhould be equal to a given {quare, this would be done by 
finding G; and then becaufe the fum of the fquares of the 
perpendiculars is given, and has a given ratio to the fquare of 
DG, DG will be given, and confequently the point D. This 
is alfo true, whatever be the number of the lines. 
28. Tue connection of the Porifms with the impoffible cafes. 
of thefe problems, is abundantly evident; the point L being ~ 
that from which, if perpendiculars be drawn to AC. and CB, 
the fum of their fquares is the leaft poffible. For fince (fig. 6.) 
DF?+DE?:DG?::LO?+LM :LG*, and fince LG is lefs than DG, 
LO?-++L™M? muft be lefs than DF?+ DE. 
‘Hence alfo a point Q may be found, from which, if per- 
pendiculars be drawn to the fides of the triangle ABC, the fum 
of the fquares of thefe perpendiculars is lefs than the fum of 
the fquares of the perpendiculars drawn to the fides of the tri- 
angle from any other point. 
For if ad (fig. 8.) be any line drawn parallel to AB, and if 
“it be divided in a, fo that aA may be to Ad in the duplicate ra- 
tio of aC to Cd, or of AC to CB, then of all the points 
in the line ad, A is that from which, if perpendiculars be 
drawn to the lines AC, CB, the fum of their fquares is the 
. lealt poffible. But fince aa is to 44 as the fquare of AC to the 
fquare of BC, that is, as AL to LB, therefore the /ocus of A is 
the ftraight line LC, joining the given points Land C. The — 
point to be found therefore, or that from which perpendicu- - ; 
lars being drawn to the fides of the figure, the fum of their 
fquares is the leaft poffible, is in the ftraight line LC. 
For let g be any point on either fide of LC, and let the line 
ab be drawn through g parallel to AB, meeting LC in a, then 
the fam of the {quares of the perpendiculars from g upon AC, 
CB, is greater than the fum of the fquares of the perpendicu- 
Jars from A upon the fame lines. Therefore adding the fquare 
of the perpendicular from q, or a, on AB, to both, the fum of 
the fquares of the perpendiculars from g, will be greater than 
the: 3 
