Dr STEWART'S THEOREMS. 117 



equal to the fide of this fquare, defcribe a circle. The extremi- 

 ties X, Y, of any diameter, will be two fuch points as are re- 

 quired. For 



DA'+DB^+DC^ = EA^+EB^+ED+3.ED% (Theor. 6.). 



But EA^-I-EB^+EC^ = 3.EX% therefore 



2(DA^4-DB^+D,D) = 6(EX^+ED') = 3(DX^+DY^) 



(Prop. I.). 



THEOREM IX. Fig. IV. 



Let there beany number, m, of given points A, B, C, &c. and let 

 c, by c, &c. be given magnitudes, as many in number as there are 

 given points, two points X, Y, may be found, fuch, that if from any 

 point D there be drawn fraight lines to A, B, C, &c. and to X, Y, 



DA'+-fDB^+-^DC^&c. = f-^')(DX'+DY^). 

 This propofition follows, in the fame manner, from theor. 7. 

 Let /» be = 3. Let E be a point fuch that DA^+ — DB' + 



-DD = EA'-|-4eB'+-^EC^+(^')eDs On E as a 



/ „ L r 



centre, with the diftance EX = -/ ^^^^^j^p^^J^ — eB^ + — EC") 



defcribe a circle. The extremities X, Y, of any diameter, will 

 be two fuch points as are required. For 



DA»+4dB"+V^C^ = EA*+7EB"+tEC'+ f-^O^DS 

 and EA'+4EB"+-^Ee = (— *— )eX^ Therefore, 



2(DA' 



