58 INVESTIGATION of fome 
THEN, if radius be denoted by r, it is evident, that DP is — 
r—GN, DR =r—GF, DS =r+GH, DO =r + GM, and: 
DT =r-+ GV; and calling N the number of the fides of the 
figure, the fum of the fquares of thefe lines is N X77 + 27 X 
GH + GM+GV—GN— GF + GH +GM'+Gv + GN’ 
+ GF. But fince the angles HGN, NGM, MGF, FGV, are 
equal, and the angles at H, N, M, F, V, right ones, a-circle, ha- 
ving its diameter = GD, pafles through the points G, H, N, D, 
M, F, V, and its circumference is divided into equal parts at the 
points H, N, M, F, V. Wherefore GH +GN +GM + GF 
GD. 2 x-2-GD 
+ GV = axNx 52 =Nx Gre But DP + DR + 
2 —— GD , 
DS +DO +DT =N xr +N x. (Stewart’s Theor. 
Prop. 5-). Therefore 2r xX GH + GM + GV — GN — GE 
— o,or GN + GF = GH + GM. + GV.. Whence this propo- 
fition: If, from any point, perpendiculars be drawn to the fides: 
of any regular figure of an odd number of fides, circum{cribing: 
a circle, the fum of the parts by which thofe perpendiculars,. 
which are greater than radius, exceed it, is equal.to the fum of, 
thofe parts by which the perpendiculars, which are lefs than ra- 
dius, fall {hort of it. And this propofition is alfo.true with re-. 
gard to any regular figure, of which the number of its fides is a, 
multiple of any odd number by 2, fince the perpendiculars DF, 
DM, DN, DH, DV, &c. are the fame both in number and mag= 
nitude, in any regular figure of an.odd number of fides, and a; 
regular figure of double the’ number, of fides.. Gonfequently, in 
a hexagon, one of the three perpendiculars drawn from any point 
D to the diameters joining the oppofite, points of contact, is. 
equal 
