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angle asymptotes axes axis becomes centre chord circle circumscribing coincide common conic conjugate constant construction contained coordinates corresponding curve denote described determined diameter difference double draw drawn ellipse envelope equal equation evidently expression fact figure FITZGERALD fixed points follows four given line given points harmonic hence hyperbola infinity inscribed intersection involution joining known latter locus meet middle points obtain opposite origin pairs parallel pass pencil perpendiculars plane points of contact polar position problem Proposed prove quadrilateral Question radius reference relative respectively result segments semicircle shown sides similar similarly Solution sphere square straight line suppose taking tangent theorem third touch triangle ABC vertices whence
Page 16 - The line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of the third side.
Page 23 - There are n points in a plane, no three of which are in the same straight line with the exception of p, which are all in the same straight line; find the number of lines which result from joining them.
Page 36 - But this amounts to saying that if we join any three of the five points by a plane, and the other two by a line, the line and plane are conjugates. This statement makes no mention of the particular point taken for centre; and we conclude as before, that if five conicoids are drawn, by taking each of five points in succession for centre, and the remaining four for a self-conjugate tetrahedron, these five conicoids will be similar and similarly situated. A line and its conjugate plane cut the plane...
Page 106 - Construct a triangle, having given the base, tho vertical angle, and the length of the straight line drawn from the vertex to the base bisecting the vertical angle. 551. A, B, C are three given points in the circumference of a given circle : find a point P such that if AP, BP, CP meet the circumference at D, E, F respectively, the arcs DE, EF may be equal to given arcs. 552. Find...
Page 109 - If from the intersection of the diagonals of a quadrilateral inscribed in a circle perpendiculars be...
Page 82 - And this again is a particular case ofthat wonderful proposition, the involution of cubics: — All the cubics which pass through eight fixed points pass also through a ninth point. Finally, reciprocate the whole figure in respect of the self-conjugate circle of any of the triangles 234, &c. We thus get the locus of a point where the normal at (1) meets again a rectangular hyperbola circumscribing the quadrangle ; it is a cubic having its asymptotes parallel to the sides of 234, and with a double...
Page 63 - B6 in Q; QA meets Cc in r; and so on. Prove that, after going twice round the triangle in this way, we always come back to the same point. Show that the theorem is its own reciprocal. Find the analogous properties of a skew quadrilateral in space, and of a polygon of n sides in a plane. Solution by PnOFESSOB CAYLEY.
Page 92 - ... of the hodograph. Then Sp' is parallel to the tangent at p, which again is parallel to SP. Hence PSp' is a straight line. Also, since p belongs (by hypothesis) to a central orbit, the tangent at p' is parallel to Sp, ie, to the tangent at P. Hence the locus of p' is similar to that of P, and therefore Sp