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 



DAM-DB'+DC 2 = EA J -!-EB*+ECH-3.ED% (Theor. 6.). 



But EA*-f-EB*+EO = 3.EX% therefore 



2(DA-+DB 2 +DC 2 ) =3 6(EX a +ED 3 ) = 3(DX 2 +DY 2 ) 



(Prop. 1.). 



THEOREM IX. Fig. IV. 



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

 a, b t 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«+-fr>B'+JL do &C. = (^ C )(DXN-DY*). 

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

 Let m be = 3. Let E be a point fuch that DA 2 + — DB* -f 



-^DC* = EA*+4eB* + -^EC*+(^ c )eD*. On E as a 



centre, with the diftance EX = y/ — a -r— (EA 2 -{- — EB a -f- ~ EC 2 ) 



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

 be two fuch points as are required. For 



DA' + ^DBM-f DC* = EA*+£eBH™E(>+ (—^ED', 



and EA»+-f EB'+T-EC* = (— *— )EX*. Therefore, 



2(DA< 



