PROPERTIES of the CIRCLE, ^l 



of perpendiculars drawn from P and Q^to the fides of a circum- 

 fcribing figure of double the number of fides. 

 Thus, if the figure be a pentagon, we get 



r+Oa+r — Oa z± 2r 2 + 6rxOa 



r+Oc 3 +r — Oc = zr +6rxOc 



r+Ok +r — Ok zz 2r l + 6rxOk 

 r+Ob 3 +r — Ob 5 = 2r 3 + 6r x Ob 



. 



r+Od +r— Od — 2r* + 6rxOd 



Sum == i or+brxOa +Oc +Ok +Ob +Od =2 mi£ 1 QX— • 



J 2 



When the diameter PQ^bifecls the arcs BK., DH, or is per- 

 pendicular to one of the diameters pafling through a point of 

 contact, O k, O i vanifh, and it is then demonftrated exactly in 

 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 CMs 



c n 



— x r3 f and confequently that the fum of the cubes of thofe 



2 



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 contact A. Draw C m, B n 

 perpendicular to AG. Then G m is equal to each of the per- 

 pendiculars drawn from G to the fides touching the circle in the 

 points C and D ; and A m to each of the perpendiculars drawn 

 from A to the fame fides ; G n is equal to each of the perpendi- 

 culars drawn from G to the fides touching the circle in the 

 points B, E, and A n, to each of the perpendiculars drawn from 

 A to the fame fides. Wherefore 2 G m + 2 G n + GA (2 r) = 

 2 An + 2 Am, or r — O m + r + On + rzzAn+Amzz r — O n 

 Vol. VI P.I. F + 



