1900.] P. Dutt — Properties of the circle and analogous matters. 91 



III. — On a new method of treating the properties of the circle and 

 analogous matters. — By Promothonath Dutt, M.A., B.L. Com- 

 municated by the Natural History Secretary. 



[Received 12fcli March ; Read 4th April, 1900.] 



According to Euclid the circle is defined as a plane figure, which is 

 such that the length of any straight line drawn from a certain point 

 within the circle to the boundary is constant. 



A circle may also be defined as the locus of a point which moves so 

 that the ratio of its distances from two fixed points is constant. This 

 proposition has been proved as prop. 4 of the Theorems and Examples 

 on Bk. VI in Hall and Stevens's edition of Euclid, page 361. There the 

 proposition has been given in the following words : " Given the base of 

 a triangle and the ratio of the other two sides, to find the locus of the 

 vertex." The proof shows that the locus is a circle. I propose to take 

 this property of the circle as my starting point, and to deduce other 

 properties from it. I shall first of all proceed to show how the centre 

 of the circle can be found from the definition adopted. 

 Let A, B he two given points, and PDE 

 be the circle, so that whatever the position 

 of P, the ratio of A P to BP is constant. 



Then^ = ^ 

 BP BD 



.*. £AVB= /.BPD (prop. 3, Euc. Bk. 



VI). 

 .. AP AE 

 A1S ° BP = BE 

 .'. £BPE=£QPE (prop. A. Euc. Bk. VI). 

 .'. Z.EPD is a right angle. 

 Take C as the middle point of BE. 



By a well-known rider (Ex. 2, on prop. 32, Bk. I, Hall and Stevens 

 page 100). 



We have CP=CD = CE. 

 ■*. C is the centre of the circle. 



According to the definition adopted, it will be found that AB is 

 divided harmonically at D and E (Hall and Stevens's Geometry, 

 Example I, Bk. VI, page 360). It will appear from Example III, that 

 the straight line through B drawn at right-angles to the diameter BE 

 is the polar of A with respect to the circle. Example II shows that if 

 O be the middle point of AB, OD.OE=OBK Example I at page 233 

 (Hall and Stevens) shews that the rectangle AC, BG is equal to the 

 square on the radius. 



