1900.] P. Dutt — Properties of the circle and analogous matters. 



93 



But the circle described on AT as diameter cuts AP at N and that 

 on £!Tcuts.BPat N'. 



The proposition may be enunciated geometrically thus : — 

 If the tangent at any point P to a circle meets the line joining the 

 fixed points A, B in T and. on AT, BT are described circles cutting 

 AP, BP in N, N' respectively, then the ratio of AN to BJV' is tlie same 

 as that of AP to BP. (4). 



As an alternative and purely geometrical proof of prop. (4) the 

 following is given.* 

 Join NN' and PD. 



We have LCPB = LCPD ~ LBPD = /.GDP - £APD= LP AT). 

 .*. Z42W being the complement of LP AT) is equal to L.BPT, 

 the complement of L GPB. 



But P, N, T, N' lie on a circle. 

 .-. £N'PT= L.TNN'. 

 .-. LATN = LTNN'. 

 .'. AB is parallel to NN'. 



AN AP 

 '•' BN'~BP' 

 Let the circle described on AB as 

 diameter cut the given circle in P. 

 Then if be the middle point of AB, 

 we have OD.OE=OB*=OP 2 . 

 But OD.OE=CO*-CB* 



--=C0*-CP*. 

 .-. CO* = CP*+OP*. 

 By Euclid I, 48, LOPC is a right- 

 angle. 



Therefore, OP is the tangent at P to the given circle, and it is the 

 normal to the circle described on AB as diameter. Therefore, the 

 circle described on the line joining the fixed points as diameter cuts 

 the given circle at right angles. (5). 

 Also as proved before, PN=m. PN\ 

 .'. ON' = m, ON us PNON' is a rectangle. 



Therefore, if the circle described on the line joining the fixed points 

 cuts the given circle at P, and be the point, where the tangent at P 

 meets AB, then the distances of AP, BP from are in their inverse 

 ratio. (6). 



Fig. 3. 



# For this proof I am indebted to my nephew, Baba Benodebihari Dutt of the 

 Sanskrit College, who has given me much assistance in the composition of this 

 paper. P.D. 



