Dr STEWARTS THEOREMS. 133 



fAD*.EFM rAD\FK 

 Or, S 4 z= 3DE 4 +2^ BD'.EG* I +6J BD*.LG 2 



CD\EH*J [CD 2 .MH 



Join EK., and on it as a diameter defcribe the circle KFPENQ, 

 draw the diameter FN and divide it in O, fo that FO = 3ON, 

 and through O draw PQ^ perpendicular to FN, meeting the 

 circumference in P, Qj and join K.P, KQj, EP, EQ^ In the 

 fame manner, join EL, and on it as a diameter defcribe the 

 circle GVLRTE, draw the diameter GR, and divide it in S, 

 fo that GS zz 3SR ; through S draw TV perpendicular to GR, 

 meeting the circumference in T, V ; and join L.T, LV, ET, 

 EV. In the fame manner alfo join EM, and on it as a diame- 

 ter defcribe the circle HZEX<;M, draw the diameter HX, and 

 divide it in Y, fo that HY =. 3YX ; through Y draw Za per- 

 pendicular to HX, meeting the circumference in Z, a 5 and 

 join MZ. Ma, EZ, E*. Then, FK = EN ; LG = ER ; and 

 MH = EX. Therefore, 



rAD'.EFM fAD'.ENM fAD*| 

 S 4 = 3DE 4 -f2JBD^EG4-f-6]BD*.ER 2 Uf-f BD 4 [. But 

 lCD\EH*J lCD 2 .EX 2 J [CD 4 J 



2EF 2 +6EN 2 =8FO.ON+8E0 2 =8(OP 2 +E0 2 )rr 4 (EP 2 +EQJ. 

 In the fame manner, 2 EG 2 +6ER 2 =: 8GS.SR+8ES 2 - 

 8(TS 2 -fES a ) = 4(ET*+EV 2 ). In the fame manner alfo, 

 2EH'+6EX 2 = 8HY.YX-J-8EY 2 == 8(ZY 2 -J-EY 2 ) =: 

 4(EZ 2 +Ea 2 ), Therefore, 



fAD 2 (EP'-f-EQ^)l fAD 4 l 

 S 4 = 3 DE 4 +4<BD 2 (ET 2 +EV 2 ) -K BD 4 . Since then there 

 ICD 2 (EZ 2 +E* 2 ) J ICD 4 J 



are fix ftraight lines KP, KQ^ LT, LV, MZ, Ma, given by pofi- 

 tion, and given quantities 4AD 2 , 4AD 2 , 4BD 2 , 4BD 2 , 4CD% 

 4CD 2 , as many in number as there are lines given by pofition, 

 therefore, by Theor. 17. two ftraight lines, xy, xz^ may be" 

 found, which will be given by polition, fuch, that if from the 



point 



