STEWARTS THEOREMS. 121 



DEM. The assumed point and adjacent corner subtending an 

 angle </> at the centre, we have 



.'. 2c 4 = 4/2(1- cos <)*, 



^ /., \2 w- 



/. 2(l-cos<) 2 = ; 



/. 2c 4 = 6ft/. Q. E. D. 



The fourth proposition includes the third. The assumed 

 point may now have any position we please. Let I be its dis- 

 tance from the centre. Here we have 



c 2 = r 2 + Z 2 2rZ cos <, 

 and 2c 4 = n (/ + Z 4 + 2r 2 Z 2 ) - 4r? (r 2 + Z 2 ) 2 cos $ + 4r 2 Z 2 2 cos 2 <. 



By the values above given for 5 cos and 2 cos 2 <, this 

 becomes 



2c 4 = nr* + 4nr 2 / 2 + nZ* ................... (4), 



which is the proposition in question. 



In the fifth proposition we return to the circumscribed poly- 

 gon, and our object is to determine the sum of the fourth powers 

 of the perpendiculars. As before, 



and 



therefore 82/ = 35w.r 4 ........................... (5). 



Q. E. D. 



In the general case, when I is the distance of the assumed 

 point from the centre, 



cos <j> + 6rT 2 cos 2 < - rl 3 2 cos 3 + Z 4 2 cos 4 <, 



, 

 and . 



