A parabola \(\Gamma\) is given parametrically by \(x=at^2, y=2at\). Write down the equation satisfied by the parameters of the four points in which \(\Gamma\) is met by the circle \[ x^2+y^2+2gx+2fy+c=0, \] and hence or otherwise find the equation of the circle through the three points of \(\Gamma\) whose parameters are the roots of the equation \[ t^3+\lambda t^2+\mu t+\nu=0. \] Find also the coordinates of the centre of curvature of \(\Gamma\) at the point \((at^2, 2at)\).
Prove that the reciprocal of a conic \(\Gamma\), with respect to a circle \(\Sigma\) whose centre is a focus of \(\Gamma\), is a circle \(\Gamma'\), and identify the centre of \(\Gamma'\). Three conics \(\Gamma_1, \Gamma_2, \Gamma_3\) have a common focus and are such that \(\Gamma_2\) and \(\Gamma_3\) touch at \(P_1\), \(\Gamma_3\) and \(\Gamma_1\) touch at \(P_2\), and \(\Gamma_1\) and \(\Gamma_2\) touch at \(P_3\). Prove that the tangents at \(P_1, P_2, P_3\) meet the corresponding directrices of \(\Gamma_1, \Gamma_2, \Gamma_3\) respectively in three collinear points.
Explain what is meant by an involution of pairs of points on a line. A line \(p\) meets the sides \(BC, CA, AB\) of a triangle \(ABC\) in \(L, M, N\) respectively. If \((L, L'), (M, M'), (N, N')\) are pairs of an involution on \(p\), prove that the lines \(AL', BM', CN'\) are concurrent.
Prove that, if \(A, B, C, D\) are points on a conic \(S\) and \(A, B\) separate \(C, D\) harmonically, \(AB\) is conjugate to \(CD\) with respect to \(S\). \(A, B, C, A', B', C'\) are six points on \(S\) such that \[ \{A, A'; B, C\} = \{B, B'; C, A\} = \{C, C'; A, B\} = -1. \] Prove that \((A, A'), (B, B')\) and \((C, C')\) are pairs in an involution on \(S\).
A point on the conic \(S=y^2-zx=0\) is said to have parameter \(\theta\) if its coordinates are \((\theta^2:\theta:1)\). Find the condition that the line \(lx+my+nz=0\) should cut \(S\) in points which are conjugate with respect to the conic \(\Gamma=x^2+z^2+2yz+2xy=0\). Prove that the polar of the point with parameter \(\theta\) with respect to \(\Gamma\) satisfies this condition, and deduce that there is an infinite number of triangles inscribed in \(S\) and self-conjugate with respect to \(\Gamma\).
\(S\) and \(S'\) are two conics in a plane and \(P\) is a point in the plane. Prove that in general there is a unique point \(P^*\) which is conjugate to \(P\) with respect to both conics. If \(P\) moves on a straight line \(l\), prove that the locus of \(P^*\) is in general a conic \(\Gamma\). When \(S\) and \(S'\) are circles, determine the directions of the asymptotes of \(\Gamma\), and characterize the lines \(l\) for which \(\Gamma\) is a parabola.
Prove that the Simson line of a point D on the circumcircle of a triangle ABC bisects the join of D to the orthocentre of the triangle. Hence, or otherwise, show that the four Simson lines, obtained by taking each of A, B, C, and D in turn with the triangle formed by the other three points, are concurrent.
Prove that the inverses of two orthogonally intersecting curves are orthogonal. Show that the inverse of a circle with respect to any point as centre of inversion is either a circle or a straight line. Explain what figure is obtained by inverting a set of non-intersecting coaxal circles with respect to one of their limiting points, and show that any circle cutting all circles of the system orthogonally must pass through both limiting points.
If \(a, b, c,\) and \(d\) are any four coplanar straight lines in general position, and if O is the second point of intersection of the circumcircles of the triangles formed by the triads \(abc\) and \(abd\) respectively, prove that the circumcircles of the triangles \(acd\) and \(bcd\) also pass through O. Illustrate this result by the case of four tangents to a parabola, determining in particular the identity of the point O.
Prove that if in the tetrahedron ABCD, AB=CD and AD=BC, then AC and BD are bisected by their mutual perpendicular. Prove for a general tetrahedron that the joins of midpoints of opposite edges are concurrent, and show that the join of the midpoints of one pair of opposite edges can only be their mutual perpendicular if both the other pairs of opposite edges are equal in length, as above.