\(AB\) is a diameter and \(P\) any point of a circle \(S\). The tangent to \(S\) at \(P\) meets \(AB\) produced in \(T\). \(L\) is the mid-point of \(TP\) and \(H\) is the foot of the perpendicular from \(T\) on \(AP\) produced. Prove that \(HL\) is perpendicular to \(AB\).
Three points \(L, A, B\) are taken on a circle \(S\), and \(O\) is the mid-point of \(AB\). Prove that the tangents at \(A, B\) to the circles \(AOL, BOL\) meet in a point \(M\) of \(S\), and that the tangents at \(A, B\) to the circles \(AOM, BOM\) meet in \(L\).
Discuss briefly the process of inversion with respect to a circle. \(P_1, P_2\) are the points of contact of a common tangent of two circles \(C_1, C_2\) and \(L, M\) are the limiting points of the coaxal system determined by \(C_1, C_2\). \(P_1L\) meets \(C_1\) again in \(Q_1\), and \(P_2L\) meets \(C_2\) again in \(Q_2\). By inversion with respect to \(L\) or otherwise, prove that \(Q_1Q_2\) is a common tangent of \(C_1, C_2\).
Two variable points \(P(x,0)\) and \(P'(x',0)\) on the line \(y=0\) have their coordinates connected by the relation \[ axx' + bx + cx' + d = 0, \] where \(a, b, c, d\) are constant and \(ad \neq bc\). Show that there exist in general two points \(A\) and \(B\) on the line such that the cross-ratio \((APBP')\) has a constant value \(k\). Investigate the cases \(k=-1, 1\).
Show that, if \(P, Q, R, S\) are four concyclic points on a conic, the lines \(PQ, RS\) are equally inclined to the principal axes. Deduce that the six intersections of three circles lie on a conic if and only if the centres of the circles are collinear.
\(P\) is any point of the parabola \[ y^2=a(x-a) \] and \(O\) is the vertex of the parabola \[ y^2=4ax. \] The circle on \(OP\) as diameter meets the second parabola in three points other than \(O\). Prove that the normals at these three points meet in a point of the parabola \[ y^2 = 4a(x+a). \]
The lines joining a point \(P\) of a rectangular hyperbola \[ S \equiv xy - c^2 = 0 \] to the points \((a,a), (-a,-a)\) meet \(S\) again in \(Q, R\). Show that the locus of the pole of \(QR\) with respect to \(S\) is the hyperbola \[ (a^2x+c^2y)(c^2x+a^2y) = c^2(c^2+a^2)^2. \]
Chords of a conic \(S\) are drawn subtending a right angle at the fixed point \(P\). Prove that their envelope is a conic with \(P\) as one of its foci, and the polar of \(P\) with respect to \(S\) as the corresponding directrix.
\(S\) is the conic \[ S \equiv ax^2+2hxy+by^2+2gx+2fy+c = 0, \] and \(S'\) is the circle \[ S' \equiv x^2+y^2-r^2=0. \] Find the equation of the locus of the centres of the conics of the pencil \(S+\lambda S'=0\). Show that it meets \(S\) in the feet of the normals drawn from the centre of \(S'\) to \(S\).
The line \(lx+my+nz=0\) cuts the sides \(YZ, ZX, XY\) of the triangle of reference \(XYZ\) in the points \(L, M, N\) respectively. \(L'\) is the harmonic conjugate of \(L\) with respect to \(Y, Z\), and \(M', N'\) are similarly defined. \(O\) is the point \((p,q,r)\). \(OL'\) cuts \(M'N'\) at \(U\), and \(V, W\) are similarly defined. Prove that \(XU, YV, ZW\) are concurrent at the point whose coordinates are given by the equations \[ l(-lp+mq+nr)x = m(lp-mq+nr)y = n(lp+mq-nr)z. \]