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XIII. On a Property of the Conic Sections. By J. W. Lub- 

 bock, Esq.*. 

 IF any hexagon be circumscribed about any conic section, 

 and the opposite angles be joined, the three diagonals 

 have a common intersection!. 



This remarkable theorem was first given by M. Brianchon 

 in the 13th cahier of the Journal de VEcole Polytechnique, 

 p. 301. It was deduced by M. Brianchon through Pascal's 

 celebrated property of the inscribed hexagon, but it seems 

 desirable to obtain a direct proof of this curious theorem, 

 and in so doing I have found an equation of condition be- 

 tween the coordinates of the angles of the circumscribed 

 hexagon, upon which the property in question may be said to 

 depend. 



Let the angles of the circumscribed hexagon be denoted by 

 the figures 1, 2, 3, 4, 5, 6, as in the annexed diagram. The 

 lines 1 4, 3 6 and 2 5 have a common intersection. 



Let «, /3 be the coordinates of the intersection of the line 

 1 4 with 2 5 then, Xi,y^ being the coordinate of the point 



X. —X. 



^ ^ (^4.^1 -3/4^1) (-^2 -'%) + (■^5.^2-3/5 -^2) (-^1— -^4) 

 (j/l-3/4)('^6-«^2) + (^2-^/5) {^\—^^) 



and if a, jS are also the coordinates of the intersection of the 

 lines 1 4 and 3 6, so that the lines 1 4, 2 5, and 3 6 have 

 a common intersection, 



^ _ { ^^y^—y^ ^1) (■^3— -^e) + {^^yQ—y^Vo} (^i— -^4) 



(^1—3/4) (^3-^6) + (3/6-3/3) (^l-~'^4) 



* Communicated by the Author. ,^ — ~^^^ 



t On this subject see also a paper by Mr. Davies, Phil. Mag., First 

 Series, vol. Ixviii. p. 337.— Edit. 



G2 



