\(ABC\) is a triangle whose angle \(A\) is a right angle. Lines parallel to the opposite sides are drawn through \(B\) and \(C\), meeting the external bisector of the angle \(A\) in points \(L\) and \(P\) respectively. Show that \(BP, CL\) intersect on the perpendicular from \(A\) to the opposite side \(BC\).
\(A_1B_1C_1D_1\) and \(A_2B_2C_2D_2\) are two quadrangles such that the lines \(A_1B_1, C_1D_1, A_2B_2, C_2D_2\), meet in a point \(X\), and the lines \(B_1C_1, A_1D_1, B_2C_2, A_2D_2\), meet in a point \(Y\). A conic through \(A_1, B_1, C_1, D_1\) meets \(XY\) in two points \(P, Q\). Show, by means of involution properties or otherwise, that a conic can be drawn through the six points \(A_2, B_2, C_2, D_2, P, Q\). If \(A_1, C_1, B_2, D_2\) are collinear, show that \(A_2, C_2, B_1, D_1, X, Y\) lie on a conic.
Define conjugate points with respect to a conic, and show that the locus of points conjugate to a given point is a straight line. \(C\) is the pole of the line joining two points \(X, Y\) on a conic, and \(P\) is any other point of the conic. Show that \(PX, PY\) harmonically separate \(PC\) and the tangent at \(P\) to the conic. Interpret this result (a) when the tangent at \(P\) is the line at infinity, (b) when \(X, Y\) are points at infinity in two perpendicular directions.
Two conics \(S_1\) and \(S_2\) meet in four distinct points \(A, B, C, D\), and \(O\) is a point on the line \(AB\). The polar of \(O\) with respect to \(S_1\) meets \(S_1\) in \(X, Y\). The lines joining \(C\) to \(X, Y\) meet \(S_2\) again in \(P, Q\) respectively. Show that \(PQ\) and \(AB\) are conjugate with respect to the conic \(S_2\).
Show that two circles, in different planes, which have two points common lie on a sphere. A tetrahedron is such that the pairs of opposite edges are perpendicular. Show that the nine-point-circles of each of the four triangles which are formed by the edges of the tetrahedron lie on a sphere.
Find the equation of the circle of curvature at the origin of the parabola whose equation in rectangular Cartesian coordinates is \(y^2=4ax\). Show that the locus of the middle points of chords of the parabola which touch the circle is given by the equation \[ y^2(y^2-4ax) + 4a^2(x^2+y^2-4ax) = 0. \]
If \(lx+my+1=0\) is the equation of a straight line referred to rectangular Cartesian axes, and if \(l^2 + 2\theta lm - m^2 = 1\), find equations to determine the foci of the conic which the line touches, and show that for all values of \(\theta\) the foci lie on the rectangular hyperbola \(x^2 - y^2 = 2\).
Prove that, if \(a, b\) are positive and \(\sqrt{2} > \theta > 1\), the ellipse \(x^2/a^2+y^2/b^2=1\) meets the rectangular hyperbola \(2xy=(\theta^2-1)ab\) in four real points, two of which, \(P, Q\), are in the positive quadrant. Show that the equation of the circle on \(PQ\) as diameter is \[ x^2+y^2 - \theta ax - \theta by + \frac{1}{2}(a^2+b^2)(\theta^2-1) = 0. \] Show that, as \(\theta\) varies, this circle touches a fixed ellipse which has double contact with the director circle of the given ellipse.
The homogeneous coordinates of a point on a conic \(S\) are expressed in the parametric form \((\theta^2, \theta, 1)\). Find the pole \(P\) of the line joining the points \(A(\alpha^2, \alpha, 1)\) and \(B(\beta^2, \beta, 1)\). Show that a conic \(S'\) passes through the vertices of the triangle of reference and the points \(A, B, P\). If \(A\) and \(B\) vary in such a way that the line \(AB\) passes through the fixed point \((\xi, \eta, \zeta)\), show that the conic \(S'\) passes through the fixed point \((1/\zeta, 1/\eta, 1/\xi)\).
Points \(D, E, F\) are taken in the sides \(YZ, ZX, XY\) respectively of a triangle \(XYZ\), so that \(XD, YE, ZF\) are concurrent. A conic \(S_1\) is drawn through \(X, E, F\) touching \(YZ\) at \(D\), and conics \(S_2, S_3\) are defined similarly by cyclic interchange of letters. Show that the fourth point \(P\) of intersection (other than \(D, E, F\)) of the conics \(S_2, S_3\) lies on \(XD\), and that, if \(Q, R\) are defined similarly, then \(EF, QR, YZ\) are concurrent.