120 STEWARTS THEOREMS. 



DEM. Let the assumed point subtend at the centre an 

 angle <> from the adjacent point of contact. Then 



p = r (1 cos <), p l = r -1 1 - cos ( <f> + -J } , &c. &c. 



-cos <; 

 and by the general formula, 



w f 2 * 

 2 (1 - cos <) 3 = I (I cos $) 3 d(j> ...... n being > 3. 



Now f ^(1 - cos </>) 3 <fy = 2 4 Tain 6 dO . . . (0 = J<) , 



Jo * 



^ir ^\ Q 1 K 



and I sin 6 ^^ = g- L j 1 2.7T = -j27r; 



..S(1-C08*>'=, 



and therefore 2S 3 = 5wr s . Q. E. D. 



This is a particular case of the second proposition in which 

 the assumed point is not confined to the circumference of the 

 circle, but may have any position whatever. Let I be its dis- 

 tance from the centre ; then 



22/ = 2w s + 3/iZV ... n > 3 ............... (2). 



DEM. In this case p = r I cos <, &c. = &c. 



.-. 2/ = nr 3 - 3r 2 ? 2 cos </> + 3rZ 2 2 cos 2 < - Z 8 2 cos 8 </>. 



T2ir r2ir /*2ir 



But I cos (f>d<f> = 0, I cos 2 <pd(f> = 7r, I cos 3 <j>d<j> = 0; 



J o J o / 



and 22 8 = 2wr 3 + 3w?V. Q. E. D. 



In the third proposition, a regular w-sided polygon is in- 

 scribed in the circle, and lines c, c t , &c. are drawn from its 

 corners to a point assumed in the circumference ; then 



(3). 



