If two triangles are both self-conjugate with regard to a conic, prove that the six vertices lie on a conic. Show that the envelope of the axes of conics which touch the sides of a quadrilateral circumscribed to a circle is a parabola.
Prove that in general there are six cross-ratios of four collinear points. The pencil formed by joining a variable point to the vertices of a square is harmonic. Prove that the point lies on the circumscribed circle of the square, or on one of two circumscribed hyperbolas of eccentricity \(\sqrt{3}\).
A variable conic touches the ellipse \(ax^2 + by^2 = 1\) at points on the line \(lx+my=1\). Show that the locus of the points of contact of tangents to the conic from the centre of the ellipse is \(ax^2+by^2 = lx+my\).
Find the equation of that diameter of the conic \(ax^2+2hxy+by^2+2gx+2fy+c=0\) which is conjugate to the diameter parallel to \(y=mx\). A family of concentric circles intersect the conic. Show that the locus of the middle points of the chords of intersection is the rectangular hyperbola \[ (x-\alpha)(hx+by+f) = (y-\beta)(ax+hy+g), \] where \((\alpha, \beta)\) is the centre of the circles.
Form the general equation in homogeneous coordinates of a conic inscribed in the triangle of reference, and the corresponding tangential equation. Prove that if \(\begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix}=0\), the three lines \(x/l_1+y/m_1+z/n_1=0\), etc. touch a conic inscribed in the triangle of reference. Prove further that the three lines form a triangle inscribed in a conic in which the triangle of reference is inscribed.
Points \(P, Q\) are taken in the sides \(AB, AC\) respectively of a triangle \(ABC\), so that \(AP:AQ :: AC:AB\). Prove that the middle point of \(PQ\) lies on a line through \(A\) which makes with \(AB\) an angle equal to the angle which the median through \(A\) makes with \(AC\).
Prove that the angle at which two curves cut is equal to the angle at which their inverse curves cut at the corresponding point of intersection. \(O, P, Q\) are three non-collinear points and \(X\) is any point on \(PQ\). Invert this figure and so prove that the circles \(OPX, OQX\) cut at a constant angle.
Prove that the length of the perpendicular from the focus on a tangent to a parabola is a mean proportional between the focal distances of the point of contact and the vertex. If \(PQ\) is a focal chord of a parabola whose focus is \(S\) and vertex \(A\), show that \[ SA \cdot PQ = SP \cdot SQ. \]
The perpendiculars from the foci \(S\) and \(S'\) of an ellipse meet the tangent at \(P\) in \(Z\) and \(Z'\) respectively. Prove that \(SZ \cdot S'Z' = CB^2\), where \(C\) is the centre of the ellipse and \(B\) is the end of the minor axis. If the tangent at \(P\) meets the directrices corresponding to \(S\) and \(S'\) in \(N\) and \(N'\) respectively, prove also that \[ SZ:S'Z' :: SN:S'N'. \]
Prove that the polar reciprocal of a circle with respect to another circle is a conic section. If the radius of the first circle is \(a\) and the radius of the second circle is \(b\), prove that the latus rectum of the conic is \(2\dfrac{a^2}{b}\).