PROPERTIES of the CIRCLE. 39 



angles at O formed by the diameters pairing through the points 

 of conta<5l 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, b, e, &o ; and the fum- 

 of perpendiculars from P and O, to the fides of a circumfcribing 

 figure, touching this circle in the points a, b, e t &c. is the fame 

 whether the figure be regular or irregular.. 



Otherwise, 



Ka -\- A.c — 2r -. 4 r — Ka -f-Ac {aG) + 2 Aa x aG 



(2XYa). In like manner, 4 r 2 — Cb -f- Oct [ft) -f 2 G b 



X bh (2 X Yb~) and fo on : 4 n r z m K~a -f- Ac" + Clt -f 



Qd + B? + B/ 2 + &c -f 2 x P^ + Pb + Ye -f &o 

 whether the angles AOB, BOC, &c. at O be equal or unequal. 



In like manner, the fum of the cubes of perpendiculars drawn 

 from P and Qj:o the fides of any circumfcribing figure, regular 

 or irregular, of a given number n of fides, together with thrice 

 the folid, which has the fquares of perpendiculars from P and Q^ 

 to the diameters parting through A, B, C, &c. for its bafe, and r 

 for its altitude, is an invariable quantity ; that is, 2 n r'i -f- 6 r x 



Oa + Ob + Oe + &c. -f Pa + ¥b +?e + &c. is an inva- 

 riable quantity, being = 2«r3-J-6rx^X PO z=. 8 n r* — n x 



or 3 . For Oa + Ob + Oc + See. + Pa -f Yb + Ye -f- &c. 

 n n r\ Confequently, when AB, BC, &o 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. 



AP + BP + CP + &c. + AQ^ -J- BQ^ + CQ^ -f &c. be- 

 ing zz. 4// r% whether AB, AC, &c. be equal or unequal, is in- 

 variable* 



