336 Mr. T. S. Davies's Properties 



Cor. 1. — The proposition numbered IX. in my former pa- 

 per may be demonstrated in all its cases by binary combina- 

 tions of the different cases of Pascal's theorem. The demon- 

 stration, however, there given being upon the principles of 

 plane geometry is preferred to one in which the conic sections 

 are involved. 



Cor. 2. — Converse of Pascal's theorem : if any two lines 

 (LI, HK) intersect each other in P in either diagonal (FG) 

 of a trapezium EFDG and prolonged to cut the sides in four 

 points L, K, H, R, a conic section will pass through E, L, R, 



D, I, H. 



Cor. 3. — Conceive the lines HD, EK to meet in A', and 

 LD, EI in O; then A'C will pass through B, 



Also, if ER, LD be conceived to meet in a, and EI, DH 

 in b, then aP b will be a straight line. 



Cor. 4. — If the two lines LI, HK pass through the inter- 

 section of the diagonals of the trap'ezium, a conic section will 

 pass through H, I, K, L, and either pair of opposite angles 



E, D or F; G. 



It is obvious enough how to determine the species of the 

 curve in every case that can arise. 



Prop. II. 



Let two triangles IGL, HKM mutually intersect each 

 other in A,B,C,D,E,F, so that these points be posited in the 

 circumference of any conic section : then lines IM, KG, HL, 

 joining the opposite vertices, will intersect in the same point. 

 (See Plate IV. fig. 4.) 



Dem. — Prolong each pair of opposite sides of the hexagon, 

 ABCDEF to meet in N, O, P: then (Prop. I.) N, O, P will 

 be in a straight line. 



Again, because N, O, P are in one straight line, the lines 

 MI, KG intersect in the line HP by Prop. IX. (Trapezium, 

 Phil. Mag. No. 340.) Q. E. d. 



Cor. 1. — Prolong the alternate sides of an inscribed hexa- 

 gon till they meet : the three diagonals giving the three pairs 

 of opposite intersections will meet in a point. This is only 

 another mode of stating the above property. 



Cor. 2. — If three lines MI, GK, HL cross each other in 

 the same point P, and two triangles IGL, HKM be de- 

 scribed, the vertices of which are in these three lines, then the 

 six intersections of these two triangles are in the periphery of 

 a conic section. 



Prop. III. 



Let two triangles HID, NEM be inscribed in a conic 

 section, and by their mutual intersections form a hexagon of 



which 



