PROPERTIES of the CIRCLE. 43 



we get 4 ^- — — £■— ; and if r* be fubftituted for its equal in 



3 ? r 3 + ^o r' X O ff -f 60 r x Q"a* we 35H -f 15 r » __ 2$r * 



4 ' . 4 2 



Wherefore the fum of the cubes of perpendiculars drawn from 

 the point G to the fides of the pentagon, is equal to the fum of 

 the cubes of perpendiculars drawn from the point A to the 

 fame. 



Since 2Xr-\-Om +2X r — O n — 4 ; 3 -f 3 r 2 X zOvi — lOn 

 + 3rX2XOw 1 + 2 X 0«>2xOffl 5 -2X^« 3 —-—— and 



2 » 



2XOw+0«' = ^-, we have $r z x 2XOm — 2 X O » -f~ 

 2 X Ow' — Q»* = 4 r 3 . But lOm — 2O n=zrj therefore 

 iXOm' — On* = r*, or O^* — 6a 1 = ~ 



2 * 



If P, inflead of biferting the arc BK, be any point between 

 B and K. the fum of the cubes of perpendiculars drawn from it 

 to the fides of the circumfcribing pentagon, is equal to the fum 

 of the cubes of perpendiculars drawn from Qjo the fame. For 

 fince Of + O a + Ok = Ob + Od and O^ — O b, Od — Oa 

 and Ok, begin together, and become maxima together, Oc—Ob 

 has toOia given ratio. Let that be the ratio of m to 1. Then 

 Oc — Ob "= mxOky and Od — Oa = Oc~ Ob + Ok = 



6^ = O*' +3«X(S l xOi + 3^x0^xot-f^X 



5?. oS 3 =o5 3 -3xFT7 x ^ x0i + 3Xw+ixWx 



F 2 3 



o* 



