144 



MB. T. T. WILKINSON ON 



If, for instance, we take the values of the sides and of the 

 lines A O, BO, CO, &c., as given in the HorcB Geom. (7) 

 and (8), Diary, 1843; we have — 



«' = (r, + r,)K— r); 6^=(r,+r3) (r — r);- c* = (r,+ rj 



{r,—r). 

 . • . a' + 6* + c' = 2 (r^ r, + r^ r^+ r, r, — r r, — r r, — 



^O (1). 



Again: a^={r^—r) K— r); ^*=(r^— r) (r3— r); / = 



(rj — ^) (/"a — r)\ whence 

 2a» + 2/8' + 2y' = 2r, r, + 2r,r3+ Sr.r,— 4rr3— 



4 r r, — 4 rr^ + Q r'. 

 But2c?(D— r)= +2rr,+ 



2rr^ + 2rrj— 6r'. 

 .-. 2a" + 2^* + 2y' + 2<i(D— r) = 2(rir. + r,r3 + 



r, r,— r r,— r r,— r r.) (2). 



Now (1) and (2) are identical, and 



.-. o* + &* + c' = 2a* + 2/3* + 2y'' + 2(^(D— r); whence 

 *'the sum of the squares of the sides of any plane triangle is 

 equal to twice the sum of the squares of the lines from the 

 inscribed centre to the angular points, added to twice the 

 product of the inscribed diameter into the difference between 

 the circumscribed diameter and the inscribed radius.^* The 

 enunciation of this property is due to Mr. Henry Buckley, of 

 Delph, and would not he easy to demonstrate by the ordinary 

 forms of the Ancient Geometry. 



The properties of what the late Mr. J. H. Swale has very 

 appropriately termed *' circles of tangential ratio" appear to 

 have received a considerable share of attention from the 

 Lancashire Geometers. In the third number of the Mathe- 

 matical Companion, Mr. John Fletcher proposed for consi- 

 deration the determination of a fourth point D, when *' three 

 points A, B, C, being given, it is required that w.AD"; 

 w-BD*; ^.CD"; shall have given differences;" — and this 

 somewhat difficult problem was very elegantly constructed 

 and demonstrated by Mr. R. Nicholson, of Liverpool, and 



