INVESTIGATION of PORISMS. 187 
the fum of the fquares of the perpendiculars from a, The 
point, therefore, which makes the fum of the fquares of the per- 
pendiculars drawn from it, to. the fides of the triangle ABC, a 
minimum, is not on either fide of the line LC; it is therefore in 
the line LC. ; 
____ For the fame reafon, if AC be divided in L’, fo that AL’ is to 
t LG as the fquare of AB to the fquare of BC, and if BL’ be join- 
ed, the point to be found is in BL’, It is therefore in the point 
Q, where the lines CL and BL’ interfect one another. 
Tue point Q, in any other figure, may be found nearly in the 
fame manner. Let ABCD, for inftance, (fig. 9-) be a quadri- 
lateral figure ; let the oppofite fides, AB and DC, be produced 
till they meet in E, and let ad be drawn parallel to AB, meet- 
ing CE ine, and let A be the point in the line ad from which 
4 _ perpendiculars are drawn to the three lines BC, CD, DA, fo 
that the fum of their fquares is lefs, than if they were drawn 
_ from any other point, in the fame line ; then if weights be pla- 
_ ced at 4, a and ¢, proportional to the fquares of the fines of the 
angles Cba, baD, aeD, » is the centre of gravity of thefe 
weights. (§ 26.) Now, thefe weights having given ratios to one 
another, the /ocus of the point a, from the known properties of 
the centre of gravity, is a ftraight line La, given in pofition. 
The point to be found is, therefore, in that line. For the fame 
| reafon, it is in another ftraight line L’a’ alfo given in pofition ; 
» and therefore it is in Q, the point of their interfection. 
THERE are many other remarkable properties of this point, 
' he which appear fometimes ‘in the form of Porifms, and fome- 
times of theorems. Of the former, fome curious inftances 
\4 will be found in Dr Smatx’s Demontftrations of Dr Stew- 
3 “ART? s Theorems *. Of the latter; I thall only add one, omitting 
\- Dthe demonftration, which would lead into too long a digreflion. 
; are drawn to the fides of the triangle, fo that the fum of their 
Aquares is is the leaft poffible ; twice the area of the triangle is a 
ee mean 
* Tranf R. S. Edin. vol. ii. p. 112, &c. 
