PROPERTIES of the CIRCLE. 5$ 



It may not be unacceptable to geometers to fee the foregoing 

 conclusions in regard to regular figures circumfcribed about and 

 infcribed 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. 



Let the fides of any regular figure of an even number of fides 

 touch the circle BRETCQLS (PI. II. fig. 4O in the points >B, R r 

 E, T, C, Q^L, 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. 



If- GE, or. the radius of the circle, be denoted by r t and A a t 

 Ab, Ac, Ad, be perpendiculars to the diameters joining the 

 points. of contact, a C, a B., T Z>, S b, L c, E c, Qd, dK, are re- 

 spectively equal to the perpendiculars from the point A to the 



fides of the figure, and are alfo refpectively equal to — _, - A "- ';' 



2 r 2.r 



.» 2 « 



AT AS AL AE AO AR ^ ^ .- XT , , , 



, _ __, , , _2S=, But if N denote the num- 



2f. 2 r 2 r 2.r 2.r 2.r 



berof fides of the figure, the fum of the perpendiculars is rr 



N XT. Wherefore. AG" + AS'+aT] + &c. rr 2 NX/- 2 . 

 This is Prop. 4. Dr Stewart's Theor. 



Again, the fum of the ftjuares of the two perpendiculars 

 from A, parallel to BC, or,B a- 4- a C rr 2 r 2 + 2 X G a ; and the 

 fquaresof the two perpendiculars from A parallel to LE, or Ec 

 +j?& rr 2 r % + 2 X CTc - y *F& + S^ rr 2 r + 2 X GT; alfo 

 Kd 4. dQ^ rr 2 r 2 + 2 X Gd . Wherefore the fum of the 

 fquares of the perpendiculars drawn from the .point A to the 



iides 



