PROPERTIES of the CIRCLE. 59 



equal to the fum of the other two, and fo on ; and if, in the 

 hexagon, the point D be taken in one of the three diameters, the 

 perpendiculars drawn from it to the other two are equal. 



Again, DP 3 + OK -f DS J -f DO + DT = N X r> + 3 r 

 X GH + GM 4- GV — ON — GF + 3 r X 



GH + GM + GV + GN + GF 4. GH -f GM + GV — 



GD* 



GN — GF = N X r 3 + N X 3 r ' =±L + GH + GM + 



2 



t 



GV 3 — GN 3 — GF 3 . But fince NXr'+Nx?^ = 



^ 2 



3 



DP + DR + DS + DO + DT° GH +GM+GV = 



GN 5 + GF 3 . 



If D be in a line perpendicular from G the centre, to a dia- 

 meter drawn from any point of contact L, the odd chord GV 

 vanifhes, (V coinciding with G), and GN r: GM, GH = GF; 

 and the expreflion for the fum of the cubes of the perpendicu- 

 lars, drawn from D to the fides of the circumfcribing figure, is 



(imply NXr3-fNX3rX 



If the figure circumfcribing the circle be -a pentagon, a line 

 drawn from G, bifecting the angle QG d nearer to G. is perpendi- 

 cular to LG ; alfo, if D be in the line G d, the point M coincides 

 with D, GN = GF, GH = GV, and GM coincides with GD, and 

 twice the cube on GF or GN is equal to the three cubes on GD, 

 GH, GV, or to the cube on GD with twice the cube on GH or 

 GV ; and the difference of the cubes on GF, GV, or on GN, GH, 



H 2 is 



