114 DEMONSTRATIONS of 



Dr SiMSON, in his Reftoration of the Loci Plani, has deduced 

 them from a propofition of that book. Vid. Loc. Plan. lib. 2. 

 prop. 5. cor. I. & 3. The fecond and fom-th of Dr Stewart's 

 Theorems are particular cafes of this propofition, and are eafily 

 derived from it. 



THEOREM VII. Fig. II. 



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

 a, b, Cy &c. be given magnitudes, as many in number as there are 

 given points^ a point X may he found, fuch^ that if from A, B, C, &c. 

 there be drawn fir aight lines to any point Ti, and alfo to X the point 

 found, and if DX be joined, 



a.AE^+^.BD'+f.CD^ &c. =^J.AX^+^.BX^+f.CX^+(a+3+f)DX^ 

 Let »? be zr 3. Suppofe the point X found. Join DX ; 

 from the given points A, B, C draw AE, BF, CG perpendicular 

 to DX, and join AX, BX, CX. 



SiNCEa.AD^+^.BD'+r.CD'=j.AX'+^.BX'+f.CX^+(a+*+r) 

 WL"; and 



fl.AD^ = ^.AX^+^.DX"— 2tf.DX. XE, and 

 3.BD^ = ^.BX^+^.DX^+23.DX. XF, and 

 c.CD- — f.CX'+f.DX^+2f.DX.XGj or 

 a.AD^+^.BD^+r.CD^ = a.AX^+^.BX^+f.CX^+(«+3+f)DX'-|- 

 2DX (— fl.XE+(^.XF+f.XG) ; a.XE muft be equal, and in the 

 oppofite diredlion to ^.XF+f.XG. 



This vi\\\ be effecfted by the following conftrudlion : 

 Join AB, and divide it in H, fo that ^.BH = a.AH; that is, 

 make AH : BH zz b:a, and join HC, and divide it in X, fo 

 that HX : CX = f : a-\-b ; or ia-\-b) HX = f .CX. Then X will 

 be the point required. 



From H draw to DX, the perpendicular HK. 



Since 



