...the curve 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 inscribed in a conic section be produced to meet, the three points of section will be in a straight line ; required a demonstration...

...the arrangement of this chapter the foundation of Pascal's theorem, next in the series. PROP. XIV. The three pairs of opposite sides of a hexagon inscribed in a conic section, being produced to meet, the three points of intersection will all be situated in one straight...

...which shows that K, H, G are in the same straight line. We have therefore the following theorem : If the three pairs of opposite sides of a hexagon inscribed in a conic section be produced to meet, the three points of intersection mill range in the same straight line....

...совА — 2m coeB—2n cos(7= 0. July, 1861. PASCAL'S THEOREM. By HW CHALLIS. rrilK intersections of opposite sides of a hexagon inscribed in a conic lie on one straight line. Let P, Q (fig. 16) be intersections of two pairs of opposite aides. Suppose F to move along the conic,...

...y, 8 and p, a ; shall lie in one plane ; or, making the statement for any plane section of the cone, the points of intersection of the three pairs of opposite sides, of a hexagon inscribed in a conic, lie in one straight line. EXAMPLES TO CHAPTER VII. 1. On the vector of a point P in the plane Sap = 1 a...

...y,b and p, a ; shall lie in one plane ; or, making the statement for any plane section of the cone, the points of intersection of the three pairs of opposite sides, of a hexagon inscribed in a conic, lie in one straight line. EXAMPLES TO CHAPTER VII. ^ 1. On the vector of a point P in the plane Sap = 1...

...point (a/SyV which means the point whose trilincar co-ordinates are a, ft 7. 336. Pascal's Theorem, The three pairs of opposite sides of a hexagon inscribed in a conic intersect in points which all lie in one straight line. Let L = 0, M=0, N=0, R = 0, S=0, T = 0, be...

...and p, a ; shall lie in one plane ; or, making the statement for any plane section of the cone, 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...

...with one anotlier can be projected into concentric circles. Ex. 3. The three points of intersection of opposite sides of a hexagon inscribed in a conic lie on a straight line. [Pascal's Theorem.] Project the conic into a circle, and the line joining the points...

...self-corresponding points (when such exist), we have only to construct the straight line s which passes through the points of intersection of the three pairs of opposite sides of the hexagon AB'CA'BC' (Figs. 98, 134, 135). The self-corresponding points will then Fig. 137. be the...