PROPERTIES of the CIRCLE. 53 
Ir may not be unacceptable to geometers to fee the foregoing 
conclufions in regard to regular figures circumf{cribed about and 
inferibed in a circle, derived’ by making ufe of one point only, 
inftead of two, either in or not in. the ‘circumference, which is: 
eafily effected in the following manner. 
Lew the fides of any regular figure of an even numberof fides: 
touch the’circlé BRETCQES (PI. ff. fig. 4s) in the points B, R, 
E, T, Cc, QL, S, and let DN, DH, DM, DV, be perpendiculars 
from the point}D to the.diameters joining the points of contact ; 
and from the points of contact let chords be drawn to any point 
A in the circumference. 
Ir GE, or the-radius of the circle, be denoted by 7} and A 4, 
_ Ab, Ac, Ad, be perpendiculars to the diameters joining the 
points. of contaé, 4C, 4B, 14,84, Lc, Ec, Qd, dR, are re- 
fpectively equal to the perpendiculars from the point A to the 
nay ; é 
fides of the figure, and are alfo refpectively equal to 
— 3) at 
ee. 
nate ; Py ee Cs 
ery et) 3 pouty pt 213 to cn 
AE =a. AL eal AQ, AR But if N-denote the num-. 
Ba 2%yyar TRF SK. At) : 
ber’of fides of the figure; the fum of ‘the perpendiculars ae 
Nx “Wherefore-AG + AB. +JAT. +, &c. =. 2 N-K:7% 
This is Prop. 4. Dr STEWART’S Theor. yp - 
“AGAIN, the fim of ‘the fquares of the ‘two! perpendiculars. 
bas ,2tisq to sedans ose ; Le ee 
from A, parallel to BC, or Ba +40 = 27° +2 Ga andthe 
fquares:of» the two-perpendiculars from A: parallel to LE, or Ee 
STE ES PPE REP ETO 4s SH 2P Ha xT} alfo 
Rd 4dQ = 2r'+2XxGd. Wherefore the furn of the. 
fquares of ‘the perpendiculars drawn from the .point A to the 
fides : 
