The point \(K\) is the other end of the diameter through \(A\) of the circumcircle of the triangle \(ABC\). \(CK\) meets \(AB\) in \(S\), and \(BK\) meets \(AC\) in \(T\). Prove that \(ST\) is parallel to the tangent at \(A\) to the circumcircle, and that the tangents at \(B\) and \(C\) to the circumcircle meet on \(ST\).
One of the limiting points of a system of coaxal circles is \(L\), and the circle of the system through a point \(P\) meets the line \(LP\) again in \(Q\). Show that, if \(P\) describes a straight line, \(Q\) also describes a straight line. \par Deduce a theorem concerning confocal conics by reciprocation with respect to a circle having its centre at \(L\).
\(O, A, B, C\) are four fixed points on a conic. A variable line through \(O\) meets the sides of the triangle \(ABC\) in \(X, Y\) and \(Z\), and meets the conic again at \(P\). Prove that the cross-ratio \((PXYZ)\) is constant. \par Deduce that if two triangles are inscribed in a conic, their six sides touch a conic.
Pappus's theorem states that if \(A, B, C\) and \(A', B', C'\) are two sets of three collinear points in the same plane, the points \((BC', B'C), (CA', C'A), (AB', A'B)\) are collinear. State the dual theorem and prove it without appealing to the principle of duality.
A parabola \(S\) touches the sides \(BC, CA, AB\) of a triangle \(ABC\) at \(L, M\) and \(N\). \(BM\) meets \(CN\) in \(H\). Prove that the polar of \(H\) with respect to \(S\) passes through the centroid of the triangle \(ABC\).
Prove that the feet of the four normals from \((\xi, \eta)\) to the ellipse \[ S \equiv x^2/a^2 + y^2/b^2 - 1 = 0 \] lie on the rectangular hyperbola \[ (a^2-b^2)xy + b^2\eta x - a^2\xi y = 0. \] Deduce that the normals to \(S\) at the four points where it is cut by the lines \[ lx/a + my/b - 1 = 0, \quad x/(la) + y/(mb) + 1 = 0 \] are concurrent. \par Hence show that, if the normals at three points \(L, M, N\) of \(S\) meet on \(S\) and \(L\) is the point \((a\cos\theta, b\sin\theta)\), the equation of \(MN\) is \[ x \cos\theta/a^3 - y\sin\theta/b^3 + 1/(a^2-b^2) = 0. \]
A chord \(PQ\) is normal to a rectangular hyperbola \(S\) at \(P\), and another chord \(LM\) is drawn parallel to \(PQ\). Show that \(PL\) and \(QM\) meet on the diameter perpendicular to \(CP\), where \(C\) is the centre of \(S\).
\(POP'\), \(QOQ'\) and \(ROR'\) are three concurrent chords of a conic \(S\), and \(X\) is any other point of \(S\). \(QR, XP'\) meet in \(L\); \(RP, XQ'\) in \(M\); \(PQ, XR'\) in \(N\). Prove that \(L, M, N\) lie on a line through \(O\).
\(A, B, C\) are three fixed points in the plane of a conic \(S\), and \(M\) is a variable point of \(S\). \(AM\) meets \(S\) again in \(N\), and \(BN\) meets \(S\) again in \(L\). Prove that if, for all positions of \(M\) on \(S\), the points \(C, L, M\) are collinear, the triangle \(ABC\) is self-conjugate with respect to \(S\).
The straight line \[ l \equiv \alpha x + \beta y + \gamma z = 0 \] meets the sides \(BC, CA, AB\) of the triangle of reference in \(A_1, B_1, C_1\). \(A'\) is the harmonic conjugate of \(A_1\) with respect to \(B\) and \(C\), and \(B', C'\) are similarly defined. Show that \(AA', BB', CC'\) meet in a point \(O\) and find the equation of the conic \(S\) through \(A, B, C\) for which \(O\) and \(l\) are pole and polar. \par Prove further that, if \(l\) touches the conic \[ x^2+y^2+z^2-2yz-2zx-2xy=0, \] the conic \(S\) passes through the point \((1,1,1)\).