PROPERTIES ofthe CIRCLE. | 40 
of perpendiculars drawn from P and Q to the fides of a circum- 
fcribing figure of double the number of fides. 
Tuus, if the figure be a pentagon, we get 
(30a 47-08 = 27° 4 6rx Oa 
HOPtr_Oe 25 + Ore Oe 
r+Ok +7—O8 = ars a. 6r x OF 
r4Ob° +708 = ates 67 x OF, 
FOr +7—Od° = 204 6rxOP 
_—z 2 —_—2 
Sum = = tor+6rxOa oe +O% +08 +Od SagrsroxS 
WHEN the diameter PO bile the'arcs BK, DH, or is per- 
pendicular to one of the diameters pafling through a point of 
contaet, O%, O7 vanifh, and it is then demonftrated exa@ly im 
the fame way as in figures of an even number of fides, that the 
fum of the cubes of perpendiculars drawn from either P or Q is 
are r3, and confequently that the fum of the cubes of thofe 
drawn from P, is equal to the fum of thofe drawn from Q. But 
let the figure be a pentagon, and let the diameter AG be perpen- 
dicular to any fide in the point of conta A. Draw Cm, Bn 
perpendicular to AG. Then G7 is equal to each’ of the per- 
pendiculars drawn from G to the fides touching the circle ‘in the 
points C and D; and Am to each of the perpendiculars drawn 
from A to the fame fides; G is equal to each of the perpendi- 
culars drawn from’G to the fides touching the circle in the 
points B, E, and Az, to each of thé perpendiculars drawn from 
A to the fame fides.’ Wherefore 2Gm+2Gn+GA(2r)= 
2An+2Am, orr—Om+r+On4+r=2An+Amar—On 
_ Vor. VI.—P. I. i + 
