DEMONSTRATIONS of, &c. 113 



THEOREM VI. Fig. I. 



Let there be any number, m, of given points A, B, C, &g. a point X 

 may be found, fuch, that if from A, B, C, &c. there be drawn 

 ftraight lines to any point D, and to the point X found, and if DX 

 be joined, 



AD'+BD^+CD^ &c. = AX^+BX^+CX' &c. +ff2DX^ 

 Let »? be = 3. 



Suppose the point X found, join DX, from the given points 

 A, B, C draw AE, BF, CG perpendicular to DX, and join AX, 

 BX, ex. 

 Since AD^+BD'+CD^ = AX^+BX^+CX'+3DX% and 

 AD» = AX^+DX^— 2DX. XE, and 

 BD^ = BX^+DX^+ 2DX. XF, and 

 CD^ = CX^+DX'+ 2DX. XG, the point X in the line 

 DX muft be fo taken, that the part EX, intercepted between it 

 and AE the perpendicular from the point A, be equal to FX 

 and GX, the fum of the parts intercepted between it and the 

 perpendiculars BF and CG, from B and C ; and the parts FX, 

 GX muft be in the oppofite diredlion to EX. 



This will be efFeded by the following conftrudion : 

 Join AB, and bifedt it in H ; and join HC, and divide it in 

 X, fo that CX =: 2HX ; X will be the point required. 

 From H draw to DX the perpendicular HK. 

 Since AH = BH, we Ihall have EK - FK ; and fince 

 CX = 2HX, we Ihall alfo have GX = 2KX. Therefore fince 

 FX = FK— KX, and 

 GX = 2KX 



FX+GX = FK+KX = EK+KX = EX, and 

 — 2DX. XE+2DX. XF+2DX. XG = o. 

 The point X thus found is the centre of gravity of the three- 

 points A, B, C. This propofition, and that which follows, are 

 well known, and are given here only for the fake of order. 

 Vol. IL p Dr 



