54 INVESTIGATION of fomc 



fides of the figure, is = N x r % -f 2 X Ga -f- Gb + Gc + Gd\ 



But fince the angles G a A, G £ A, G c A, G d A, are right ones 

 a circle pafTes through the points A, a, b, G, c, d. having GA 

 for its diameter. And becaufe the angles CGT, QGG, LGQ» 

 are equal, this circle is equally divided by the points a, b, c, d. 

 Confequently the fquares of the chords drawn from thefe 



points to the point A, are together — N X ; that is, Ac (Ga ) 



z 



+ Ka (GO + Ad {Gb) + Ab (Gd) = N X — = 



4 



NX—. Wherefore the fum of the fquares of the perpendicu- 

 lars drawn from A to the fides of the figure, is = N X r* + 2 N 



, 2 



X- -NX - — • But the fum of the fquares of thefe perpen- 



AC 4 , AB 4 L AT 4 , AS* AL* , a¥* , AQ 4 

 diculars is = _ + — + -^- + — , + _ + — + -^ 



, AR 4 Therefore AG 4 + AB 4 +, &c. s= N X 6r4 = N Xar 1 



X 3 r % zz AC 2 -f- AB -f , &c. X 3 r z . Whence this propofition : 



If a circle be divided into any even number of parts, and 

 from the points of divifion chords be drawn to any point in the 

 circumference, the fum of the fourth powers of thefe chords is 

 equal to the fum of their fquares, multiplied, by thrice the fquare 

 of radius. 



Wherefore 



