n6 DEMONSTRATIONS of 



a.AO+b.W -{a+b) AX.BX-f (a+b) CX* (prop. I.), and 

 2j.AM 2 -f 2^.BN J -r- (a+b) AX.BX are given fpaces. Therefore 

 2(CD*+CE4-CP-fCGM-CH*-fCK 2 -f-CL 2 ) = (a+b) CX 2 -f-A% 

 A 2 being a given fpace. 



Cor. 2. Let any number of femicircles be given by pofition, 

 and let an equilateral figure be defcribed about every femicircle, 

 a point may be found, fuch, that if from any point there be 

 drawn perpendiculars to all the fides of the figures, and a ftraight 

 line to the point found, twice the fum of the fquares of the 

 perpendiculars will be equal to the multiple of the fquare of the 

 line drawn to the point found, by the number of all the fides, 

 together with a given fpace. 



Cor. 3. Let any number of circles and femicircles be given 

 by pofition, and about every circle and femicircle let an equila- 

 teral figure be defcribed, a point may be found, fuch, that if 

 from any point there be drawn perpendiculars to all the fides of 

 the figures, and a ftraight line to the point found, twice the fum 

 of the fquares of the perpendiculars will be equal to the multi- 

 ple of the line drawn to the point found, by the number of the 

 fides, together with a given fpace. 



THEOREM VIII. Fig. IV. 



Let there le any number ; m, of given points A, B, C, &c. two 

 points X, Y, may be found, fuch y that if from any point D Jiraight 

 lines be drawn to A, B, C, &c and to X, Y, 



2(DAM- DB'-f-DG*) = W2(DX*+DY 2 ), 



This propofition follows directly from theor. 6. Let m rr 3, and 



let E be the centre of gravity of the three points A, B, C. The 



fquares of EA, EB, EC, are given, and confequently a fquare zz 



i(EA*-{-EB*-f EC J ) may be found. On E with the diftance EX 



equal 



