Two lines in a plane meet in \(K\). Prove that successive reflection in the two lines is equivalent to a rotation about \(K\). The four points \(A,B,C,D\) lie on a circle of radius \(r\). Prove that successive reflection in \(AB, BC, CD, DA\) is equivalent to a translation through a distance \(AC \cdot BD/r\).
Prove that a point lies on the circumcircle of a triangle if and only if the feet of the perpendiculars from the point to the sides of the triangle are collinear. Four lines in a plane form four triangles. Prove that the four circumcircles have a common point \(Q\). Prove also that the four circumcentres lie on a circle through \(Q\).
Prove that the four points \(A,B,C,D\) on a conic \(S\) are concyclic if and only if \(AB, CD\) are equally inclined to a principal axis of \(S\). Deduce that the six points of intersection of three circles lie on a conic if and only if the centres of the circles are collinear.
Define the cross-ratio of four points on a line, and harmonic conjugacy between two pairs of points on a line. \(A,B,C,D\) are four collinear points; \(A'\) is the harmonic conjugate of \(D\) with respect to \(B,C\); \(B'\) is the harmonic conjugate of \(D\) with respect to \(C,A\); \(C'\) is the harmonic conjugate of \(D\) with respect to \(A,B\). Prove that the cross-ratios \((ABCD)\) and \((A'B'C'D)\) are equal.
If \(\Sigma\) is a central conic and \(S\) a focus of \(\Sigma\), prove that the reciprocal of \(\Sigma\) with respect to a circle with centre \(S\) is a circle and identify the image under reciprocation of the minor axis of \(\Sigma\). [Candidates may assume, if they wish, the formula for a conic in polar co-ordinates.] The point \(S\) is a focus both of a central conic \(\Sigma\) with minor axis \(l\) and of a parabola \(\Sigma'\) which touches \(l\); a common tangent to \(\Sigma\) and \(\Sigma'\) touches \(\Sigma\) at \(P\) and \(\Sigma'\) at \(P'\). Prove that \(SP, SP'\) are conjugate lines with respect to each of \(\Sigma\) and \(\Sigma'\).
The equations of the sides of a triangle referred to rectangular Cartesian axes are \[ u_i = a_ix+b_iy+c_i=0 \quad (i=1,2,3); \] prove that the orthocentre of the triangle is given by \[ \lambda_1 u_1 = \lambda_2 u_2 = \lambda_3 u_3, \] where \[ \lambda_1 = a_2a_3+b_2b_3, \quad \lambda_2 = a_3a_1+b_3b_1, \quad \lambda_3 = a_1a_2+b_1b_2. \] If \(D_1, D_2, D_3\) are the feet of the perpendiculars from the vertices of the triangle to the sides \(u_1, u_2, u_3\) respectively, prove that the intersections of \(u_1\) and \(D_2D_3\), of \(u_2\) and \(D_3D_1\), and of \(u_3\) and \(D_1D_2\) lie on the line \[ \lambda_1 u_1 + \lambda_2 u_2 + \lambda_3 u_3 = 0. \]
Find the centre, asymptotes, length of the real principal axis and real foci of the hyperbola whose equation in rectangular Cartesian co-ordinates is \[ 32x^2+52xy-7y^2+12x+66y+153=0. \]
Four distinct points lie on a rectangular hyperbola: prove that in general there are two parabolas through the four points and that their axes are equally inclined to the asymptotes of the hyperbola. Prove further that, if the four points are concyclic, the axis of each parabola is parallel to one of the axes of the hyperbola.
The lines joining a point \(P\) to the vertices \(X,Y,Z\) of a triangle \(XYZ\) meet the opposite sides in \(L,M,N\) respectively; \(L'\) is the harmonic conjugate of \(L\) with respect to \(Y,Z\) and \(M', N'\) are similarly defined. Prove that \(L', M', N'\) lie on a line \(p\). Find the envelope of \(p\) (i) when \(P\) moves on a fixed conic through \(X,Y,Z\), (ii) when \(P\) moves on a fixed line in general position.
Prove that with a suitable choice of homogeneous co-ordinates the parametric equation of a conic can be taken in the form \[ x:y:z = t^2:t:1. \] The polar line of the point \((t^2, t, 1)\) with respect to the conic \(S=(y+z)^2+2zx=0\) meets the conic \(S'=y^2-zx=0\) in \(P\) and \(Q\); prove that \(P\) and \(Q\) are conjugate with respect to \(S\) and deduce that an infinite number of triangles self-polar with respect to \(S\) can be inscribed in \(S'\). Prove that all such triangles are circumscribed to the conic \[ y^2-3z^2-2yz-4zx=0. \]