INVESTIGATION of PORISMS. 187 



the fum of the fquares of the perpendiculars from X. 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 LG ; it is therefore in 

 the line LC. 



For the fame reafon, if AC be divided in I/, fo that AL' is to 

 L/C 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. 



The 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 ab be drawn parallel to AB, meet- 

 ing CE in e, and let X be the point in the line ab from which 

 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 £, a and e, proportional to the fquares of the fines of the 

 angles Cba, baD, aeD, X is the centre of gravity of thefe 

 weights. (§ 26.) Now, thefe weights having given ratios to one 

 another, the locus of the point X, from the known properties of 

 the centre of gravity, is a flraight line L X, given in pofition. 

 The point to be found is, therefore, in that line. For the fame 

 reafon, it is in another flraight line L'x' alfo given in pofition ; 

 and therefore it is in Q. the point of their interfection. 



There are many other remarkable properties of this point, 

 which appear fometimes in the form of Porifms, and fome- 

 times of theorems. Of the former, fome curious inflances 

 will be found in D'r Small's Demonftrations of Dr Stew- 

 art's Theorems *. Of the latter, I mail only add one, omitting 

 the demonftration, which would lead into too long a digreffion. 



If Q^be the point in a triangle from which perpendiculars 

 are drawn to the fides of the triangle, fo that the fum of their 

 fquares is the leaft poffible ; twice the area of the triangle is a 



A a 2 mean 



* Tranf. R. S. Edin. vol. ii. p. 112, &c. 



