of Pascal's Hexagramme Mystique. 337 



which the diagonals are OL, FG, PK : these will pass through 

 the same point Q. (See fig. 5.) 



Dent. — Since (Prop. I. case 32. is suited to the figure) the 

 points F, R, Q are in a straight line, it follows (by Prop. VII. 

 Properties of Trapezium, Phil. Mag. No. 340.) that OL, KP 

 also intersect in FG. q. e. d. 



Cor. 1. — Conversely, if the angles of a hexagon be situated 

 upon the lines which pass through the same point, the al- 

 ternate sides being prolonged will meet in six points through 

 which a conic section will pass. 



Prop. IV. 



The three diagonals HL, IM, RK of a hexagon circum- 

 scribing a conic section pass through the same point S. (See 



fig. 6.) 



Dem. — Draw BA, DE to meet in O, AF, CD to meet in 

 P, and FE, BC to meet in Q: then (Prop. I.) Q,P,Q are in 

 the same straight line. 



Now, it is well known that whilst the locus of O is a straight 

 line, the diagonal HL passes through a certain point S. And 

 hence it follows, since P.Q are in the same straight line with 

 O, the diagonals IM and RK also pass through the same 

 point S. q. e. d. 



Cor. 1. — If the three diagonals of a hexagon pass through 

 the same point, a conic section which touches five of the sides 

 will also touch the sixth. 



Cor. 2. — If two triangles touch a conic section the lines 

 joining the three pairs of opposite intersection will pass through 

 the same point. 



Cor. 3. — If two points of contact coalesce as A and F, the 

 point R will also coalesce with them, and the figure is converted 

 into a pentagon. From which we derive an easy method of 

 describing a trajectory to touch five lines given in position. 

 (Newt. Princip. lib. i. sect. 5. pr. 27.) Draw the diagonals 

 (fig. 7.) LI, HD to intersect in S; the line KS being drawn 

 to cut the opposite side in R, one of the points of contact. In 

 the same manner may the other points of contact be found, 

 and the section may then be described as usual. 



Cor. 4. — Let HGNL be a trapezium circumscribing a 

 conic section; and IK, RM tangents, each cutting one of the 

 triangles into which the diagonal HL divides the triangles 

 in I, K, R, M ; then RK, IM will intersect in HL. 



Cor. 5. — The lines UV, NG intersect also in the diagonal 

 MI ; and the same of the others. This corollary is, properly 

 speaking, a case of the fourth : viz. when the •prolongations of 

 the sides of the trapezium are cut by the two tangents. 



Vol. 68. No. 343. Nov. 1826. 2 U Pkop. 



