+ 
PROPERTIES of the CIRCLE. 39 
angles at O formed by the diameters: paffing through the points 
of contact A, B, C, &c. be equal or unequal, or whether the cir- 
cumference of the circle, of which the diameter is PO, be equal- 
ly or unequally divided in the points a, 4, ¢, &c.; and the fum 
of perpendiculars from P and O, to the fides of a circumfcribing 
figure, touching this circle in the points a, 4, e; &c. is'the fame 
whether the figure’ be regular or irregular. 
OTHERWISE, 
AatAca2r:4r Aad + Ac (@G) + 2AaxaG 
(2xPa). In like manner, 47? = Ch + Cd (37) + 2Céb 
x bb (2X PS) and fo on: 4nr° = Aa + Ac + Gh + 
Ca 4 Be + BF + &e. + 2xPa 4 Ph 4 Pe + &e. 
whether the angles AOB, BOC, &c. at O be equal or unequal. 
In like manner, the fum of the cubes of perpendiculars drawn 
froni Prand Q to the fides of any circumfcribing figure, regular 
or irregular, of a given number 7 of fides, together with thrice 
_the folid, which has the {quares of perpendiculars from P and Q. 
to the diameters pafling through A, B, C, &c. for its bafe, and r 
for its altitude, is an invariable quantity ; that is, 2773 + 67 x 
Oa + 0b.4+0¢ + &.+ Pa +PS +P. + &c. is an inva- 
riable quantity, being = 2ur3+ 6r xX PO' = 8273 =nx 
Oe ae —2 —2 ———_* eee eee 
gr. For O2 + Ob + Oc + &c. + Pa +Pb + Pe + &e. 
=anr. Confequently, when AB, BC, &c. are unequal among 
themfelves, the fum of the fquares, cubes, &c. of perpendiculars 
drawn from P and Q to lines touching the circle in the points: 
A, B, C, &c. is not invariable. bygnl 
AP + BP + CP + &c. + AQ +BQ + OQ + &e. be- 
ing = 47” r’, whether AB, AC, &c. be equal or unequal, is in-. 
variable. 
