Dr StEWART's THEOREMS. 119 



This propofition follows diredlly from the firft cafe of theor. 2. 

 Let m be =r 3. The points B, C, D, are in a circle of which 

 EA is the diameter, and therefore (lemma 2.) the arches BC, CD, 

 DB, are equal. Therefore, 



2(EB^-fEC=+ED') = 4.3.R' = 3.EA' t- 



Cor. I. If AB, AC, AD, interfedl one another in a given point 

 A, and make all the angles round it equal ; and if from any 

 point E there be drawn perpendiculars to AB, AC, AD ; and if 

 the fum of the fquares of the perpendiculars be equal to a 

 given fpace, the point E will be in the circumference of a given 

 circle. 



The double of the given fpace is ot.AE% therefore AE is 

 given in magnitude, and fince the point A is given, the point 

 E is in the circumference of a given circle. 



* Cor. 2. If the circumference of a circle FGH, of which 

 the radius is R, be divided into m number of equal parts, by 

 the femidiameters AF, AG, AH, &c. making with any diameter 

 EN the angles FAE, CAN, HAE, &c. twice the fum of the 

 fquares of the fines, or cofines of thefe angles will be = mK^. 



Let «2 be n 3. 



FK = EB J GL = EC i HM = ED. Therefore aCFK^' + 

 GL'+HM')=: 3EA=' = 3R^ In the fame manner, AK = AB ; 

 AL = AC J AM = AD. Therefore 2(AK*+AL*-|-AM0 = 

 3.EA= = 3R^ 



* LEMMA III. Fig. VII. 



Lei there be a figure ABCD given in fpecies infcribed in a circle^ 

 the Jiraight line EH drawn from E, the centre of the circle^ to H, 



the 



t R is the radius of the circle ABC. 



