Dr STEWARTS THEOREMS. 125 



and for the fame reafon OP m %jf*. Therefore pq- = Or = 

 \mn ~ ±Kk. But the angle OxN is given, for it is equal to 

 £EF; and fince Os is given, and Nx — Nq—sq, NO is alfo 

 given. But (Fig. 10. No. 1.) fince the lines CL, CM, inter- 

 feding in the point C, are given by pofition, and from the 

 point E there are drawn to them the perpendiculars EL, EM, 

 and LM is joined, and bifedted in O, and from O there is 

 drawn a ftraight line ON given both by pofition and magni- 

 tude, and YNZ is drawn through N to meet EL, EM in Y, Z, 

 and fo as to be bifected in N, and from Y and Z, YX, ZX are 

 drawn parallel to CL, CM ; therefore, by Lemma 4. YX, ZX 

 are given by pofition ; and confequently the point X, of their 

 interferon is given, and therefore alfo XP, XQ^ XR, XS. 



But EY, EZ, are perpendicular to XY, XZ; and it has been 

 proved that 2(EF 2 +EG 2 +EH 2 +EK 2 ) z=. 4 (E Y 2 +EZ 2 ) -f 

 2(XP 2 -fXQj4-XR 2 +XS a ), and thefe four laft fquares- are 

 given. Therefore XY, XZ, are the two lines required to be 

 found, and 2 (EF 2 -fEG 2 +EH 2 -fEK 2 ) = 4 (EY 2 +£Z 2 )+A 2 . 



The point X, found in this proportion, is the centre of gra- 

 vity of the four points P, Q»_ R, S, where perpendiculars, drawn 

 from it, meet the four lines given by pofition. It is alfo a 

 point, fuch, that the fum of the fquares of the perpendiculars 

 drawn from it, to the lines given by pofition, is a minimum. 



Cor. If the ftraight lines (Fig. 11.) AB, BC, CA, be fo fi- 

 tuated as to form an equilateral figure about a circle, or a femi- 

 circle ; or if the number of the lines given by pofition be even, 

 and every two and two interfecl each other at right angles, the 

 two lines XY, XZ, that may be found, will interfect each other 

 at right angles. 



Let the lines AB, BC, CA, that are given by pofition, form 

 an equilateral triangle. Let X be the point in that triangle, 

 which is the centre of gravity of the three points K, L, M, 



where 



