338 Mr. T. S. Davies's on Pascal's Hexagramme Mystique. 



Prop. V. 



Let two triangles OSQ, NPR intersect each other in a 

 conic section in A, B, C, D, E, F (as in Prop. II.); let ace, 

 bfd! be the triangles whose contacts with the conic section 

 are A, B, C, D, E," F, and N' s O', P', Q', R', S be their mutual 

 intersections (as in Prop. IV.); and, lastly, let n, o,p, q, r, s be 

 the intersections of the sides of the two inscribed triangles (as 

 in Prop. III.) See fig. 8. Then, 



1°. The six lines PS, P'S', ps, ad, AD, and EP will pass 

 through the same point, I ; the six NQ, N'Q', n q, be, AD, 

 and FC will pass through another point H; and the six OR, 

 O'R', or, cf, EP and FC will pass through a third point G. 



2°. The intersections of a d, be,fc will be situated in the 

 three diagonals S'F, N'Q', O'R'. 



Dem. — First, the line S'P' is one of the dingonals of a tan- 

 gential trapezium, and FC, EB the diagonals of the inscribed 

 one: therefore S'P' passes through their intersection I. 



Also by Prop. I. (cases 21. and 32. in reference to this dia- 

 gram), SP and sp pass through I. 



Again, (by Prop. IV. cor. 4.) the line a d also passes 

 through I. 



In a similar manner the other two clusters of lines are 

 proved to pass through H and G. q. e. l mo D. 



Secondly, It is clear also from Prop. IV. cor. 4. that a d,fc 

 intersect in N'Q' ; a d, b e in S'P'; and be,fc in O'R'. When 

 &c. Q. E. 2°D. 



Cor. 1. — A conic section will touch all the sides of the 

 hexagramme, n op qr s; another may be drawn to touch the 

 hexagramme NOPQRS. 



Cor. 2. — Draw NOPQRS to meet alternately; the six points 

 of section will be in a line of the second order : and the points 

 N', O', P', Q', R', S' will be in another. 



It is obvious that these processes may be extended indefi- 

 nitely within and without the given conic section ABCDEF. 



Cor. 3. — The points P, P', p are in one straight line ; and 

 the same is true of Q, Q', q ; R, R', r ; &c. 



For in this case, two angles of the hexagram have coincided 

 with two others, so that two of its sides are represented by 

 the tangents BP', CP'; whence this is a specimen of case 11th 

 of Prop. I. and the truth of the corollary is obvious. 



Cor. 4. — The points aOb are in one straight line; and the 

 same is true of b, P, c; cQd; &c. 



For it is well known that the opposite sides of the tan- 

 gential trapezium N'O'P'R' meet in the same line as the op- 

 posite sides of the inscribed trapezium ABCF. 



Cor. 



