Dr SfEWARit's THEOREMS. 119 



This proportion follows directly from the firft cafe of theor. 2. 

 Let m be 2= 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 2 -{-EC 2 +ED 2 ) - 4.3.R 2 = 3.EA 2 t- 



Cor. 1. If AB, AC, AD, interfect 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 wz.AE 2 , 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, &o making with any diameter 

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

 fquares of the fines, or cofines of thefe angles will be — mK 2 . 



Let m be = 3. 



FK = EB ; GL = EC ; HM = ED. Therefore 2(FK 2 + 

 GL 2 -f HM 2 ) = 3EA 2 = 3R 2 . In the fame manner, AK = AB; 

 AL = AC ; AM = AD. Therefore 2(AK 2 -f-AL 2 -f-AM 2 ) = 

 3.EA 2 = 3R 2 . 



* LEMMA III. Fig. VII. 



Let there be a figure ABCD given in /pecies infcribed in a circle i 

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



the 



f R is the radius of the circle ABC. 



