5 8 INVESTIGATION of fomt 



Then, if radius be denoted by r, it is evident, that DP is r= 

 r — GN, DR = r — GF, DS = r + GH, DO = r + GM, and 

 DT rz r -f- G V ; and calling N the number of the fides of the 

 figure, the fum of the fquares of thefe lines is N X ■ r* + 2 r 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, pafTes 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 



+ GV* = 2XNX^-=N x-^- But DP' + DR + 



WS + DO' 4- DT 2 = N X n + N X — • (Stewart's Theor. 



Pro r . 5.). Therefore 2 r X GH + GM + GV — GN - GF 

 = o, or GN '+ GF = GH -h 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, circumfcribing 

 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 fliort 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. Confequently, in 

 a hexagon, one of the three perpendiculars drawn from any point 

 D to the diameters joining the oppofite points of contad, is. 



equal 



