...he extended Pascal's mystic hexagram theorem, which states that the three points of intersection of pairs of opposite sides of a hexagon inscribed in a conic section are collinear (see Exercise 23.2), by showing that if the six vertices are taken in all possible orders,...

...portion, and J a3 (8 TT — 64 + 27 \/3) for the smaller portion, XV. OR PRIZE QUEST. (1703); fy PASCAL. If the three pairs of opposite sides of a hexagon...conic section be produced to meet, the three points of section will be in a straight line ; required a demonstration without the aid of perspective, or any...

...of the curre formed by the intersection of the two surfaces. XV. OR PRIZE QUEST. (1703); by PASCAL. If the three pairs of opposite sides of a hexagon...conic section be produced to meet, the three points of section will be in a straight line ; required a demonstration without the aid of perspective, or any...

...correspond as a general rule to imaginary points. 97. Pascal's theorem. The three points of intersection of the three pairs of opposite sides of a hexagon inscribed in a conic are collinear. Let A, B, C and A', B', C' be any six points on a conic. If A, A' ; B, B'; C, C' are...

...inscribed in the hyperbola; a sixth point is on OY. By a well-known theorem the points of intersection of the three pairs of opposite sides of a hexagon inscribed in a conic lie on a straight line. Join AB and produce it to cut ID produced (the ordinate through D) at E ; join...

...A'B'C' and K'L'M'. INSCRIBED AND CIRCUMSCRIBED HEXAGONS. 118 . Pascal's Theorem. The intersections of the three pairs of opposite sides of a hexagon inscribed in a conic are collinear. .---" M Fig. 92 (1). Let ABCDEF be the hexagon, and let AB, DE meet at K; BC, EF at...

...making the statement for any plane section of the cone : — In order that the points of intersection of the three pairs of opposite sides, of a hexagon inscribed in a curve, may always lie in one straight line, the curve must be a conic section. EXAMPLES TO CHAPTER...