A circle \(S\) is described on \(AB\) as diameter, and \(CD\) is any chord of \(S\). The line through \(A\) perpendicular to \(CD\) meets \(BC\) in \(C'\) and \(BD\) in \(D'\). \(C'C\) is produced to \(C''\), where \(CC''=C'C\), and \(D'D\) is produced to \(D''\), where \(DD''=D'D\). Prove that the triangles \(AC''B\) and \(ABD''\) are similar.
Two points \(P\) and \(Q\) are inverse with respect to a circle \(\Gamma\). \(P', Q'\) and \(\Gamma'\) are the inverses of \(P, Q\) and \(\Gamma\) with respect to a coplanar circle \(S\). Prove that \(P'\) and \(Q'\) are inverse points with respect to the circle \(\Gamma'\), and interpret this result when the centre of \(S\) lies on \(\Gamma\). \(A, B, C, D\) are coplanar points. \(A'\) is the inverse of \(A\) with respect to the circle \(BCD\), \(B'\) the inverse of \(B\) with respect to the circle \(CAD\), and \(C'\) the inverse of \(C\) with respect to the circle \(ABD\). By inversion with respect to \(D\) prove that the circles \(A'BC, B'CA, C'AB\) have a common point.
Find the co-ordinates of the point of intersection of the normals to the parabola \[ y^2 - 4ax = 0 \] at its intersections with the polar of the point \((\xi, \eta)\). Deduce the co-ordinates of the centre of curvature at the point \((at^2, 2at)\).
A chord \(PQ\) is normal at \(P\) to a rectangular hyperbola whose centre is \(C\). \(RS\) is another chord parallel to \(PQ\). Show that \(PR\) and \(QS\) meet on the diameter perpendicular to \(CP\).
Two parabolas have a common focus and their axes are perpendicular. Prove that the directrix of either passes through the point of contact of their common tangent with the other.
A variable conic has a fixed focus and touches each of two parallel lines. Prove that its asymptotes envelop a circle.
Four points \(A, B, C, D\) are coplanar. \(AD\) and \(BC\) meet in \(P\), \(BD\) and \(CA\) meet in \(Q\), and \(CD\) and \(AB\) meet in \(R\). \(QR\) meets \(AD\) in \(H\) and \(PQ\) meets \(AB\) in \(K\). Prove that \(HK, BD, RP\) are concurrent and that \(BH, DK, CA\) are concurrent.
Find the equation of the chord joining the points \(\theta_1\) and \(\theta_2\) of the conic \[ x:y:z = \theta^2:\theta:1. \] Show that the chords for which \[ \theta_1 = k\theta_2, \] where \(k\) is a constant, envelop a conic, and find its equation in point co-ordinates. Investigate the case \(k+1=0\).
Prove Pappus' Theorem that, if \(A_1, B_1, C_1\) and \(A_2, B_2, C_2\) are two sets of three collinear points in a plane, the points of intersection of the three pairs of lines \(B_1C_2, B_2C_1\) and \(C_1A_2, C_2A_1\) and \(A_1B_2, A_2B_1\) are collinear. State, without proof, the dual theorem. The triangles \(ABC, A'B'C'\) are in perspective from \(D\), and the triangles \(ABC, C'A'B'\) are in perspective from \(E\). Prove that the triangles \(ABC, B'C'A'\) are in perspective.
The tangents at the vertices of a triangle \(ABC\) inscribed in a conic \(S\) meet the opposite sides in \(D, E, F\). Prove that \(D, E, F\) are collinear. The line \(DEF\) meets \(S\) in \(H\) and \(K\). Prove that all conics through \(A, B, C\) which have \(H\) and \(K\) as conjugate points pass through the pole of \(DEF\) with respect to \(S\).