Dr STEWART'S THEOREMS. 133 



rAD\EFM rAD^FKM fADM 

 Or, S* = 3DE*+2^ BD^EG^ I +6\ BDM.G" 1+^-^ BD* \. 

 [CD'.EHO ICD^MH^J IcdM 



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

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

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

 circumference in P, Qj and join KP, 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 — 3SR ; through S draw TV perpendicular to GR, 

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

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

 ter defcribe the circle HZEXwM, 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; and 

 join MZ. Ma, EZ, Ea. Then, FK = EN ^ LG — ER ^ and 

 MH = EX. Therefore, 



fAD^EFM fAD^ENM fAD*| 

 S♦ = 3DE*+2JBD^EG4+6]BD^ER* 1+^ BD^^k But 

 [CD^EH-J [CD^EX0 ICD^J 



2EF'+6EN'=8FO.ON+8EO='=8(OP'+EO')=4(EP'H-EQ^). 

 In the fame manner, 2EG'+6ER^ = 8GS^R+8ES= = 

 8(TS'+ES») = 4(ET'+EV'). In the fame manner alfo, 

 2EH'+6EX» = 8HY.YX-}-8EY' = 8(ZY^+EY') = 

 4(EZ'+Ea^). Therefore, 



rAD'(EP^-j-EQ^)1 fAD*] 

 S* =3DE*+4{bD*(ET»+EV^) +VBD^k Since then there 



lCD^(EZ'+E<»0 ^ ^CD-'J 

 are fix flraight lines KP, KQ^ LT, LV, MZ, M^, given by pofi- 

 tion, and given quantities 4AD*, 4AD*, 4BD% 4BD', 4CD% 

 4CD', 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 



