Lines \(\alpha, \beta, \gamma\) are drawn through the respective vertices \(A, B, C\) of a triangle \(ABC\). Establish a necessary and sufficient condition for the concurrence of \(\alpha, \beta, \gamma\), in terms of the sines of the angles that these lines make with the sides of the triangle at its vertices. Points \(D, E, F\) are taken arbitrarily on the perpendiculars from a general point \(P\) to the sides \(BC, CA, AB\) respectively. Prove that the perpendiculars from \(A, B, C\) to \(EF, FD, DE\) respectively are concurrent.
Equilateral triangles \(BCD, CAE, ABF\) are constructed on the sides of a triangle \(ABC\) and external to this triangle. Prove that (i) the lines \(AD, BE, CF\) are concurrent; (ii) the circumcentres of the three given equilateral triangles are the vertices of another equilateral triangle.
\(A_1A_2A_3A_4\) is a tetrahedron and \(O\) is a point in general position. On each edge \(A_rA_s\) the point \(P_{rs}\) is the intersection of that edge with the plane through \(O\) and the opposite edge. Prove that the lines \(P_{13}P_{23}, P_{14}P_{24}\) and \(A_1A_2\) are concurrent at, say, \(Q_{12}\). If the corresponding point \(Q_{rs}\) is taken on each edge, prove that \(Q_{12}, Q_{13}, Q_{23}\) are collinear and that the six points \(Q_{rs}\) are coplanar.
A triangle \(ABC\) is inscribed in a circle \(\Sigma\) and circumscribed to a parabola \(\Gamma\). Prove that the focus \(S\) of \(\Gamma\) lies on \(\Sigma\). Another parabola \(\Gamma'\) with focus \(S'\) is inscribed in the triangle \(ABC\) and has its axis perpendicular to that of \(\Gamma\). Prove that \(SS'\) is a diameter of \(\Sigma\).
A parabola \(\Gamma\) is given parametrically by \(x=at^2, y=2at\). Write down the equation satisfied by the parameters of the four points in which \(\Gamma\) is met by the circle \[ x^2+y^2+2gx+2fy+c=0, \] and hence or otherwise find the equation of the circle through the three points of \(\Gamma\) whose parameters are the roots of the equation \[ t^3+\lambda t^2+\mu t+\nu=0. \] Find also the coordinates of the centre of curvature of \(\Gamma\) at the point \((at^2, 2at)\).
Prove that the reciprocal of a conic \(\Gamma\), with respect to a circle \(\Sigma\) whose centre is a focus of \(\Gamma\), is a circle \(\Gamma'\), and identify the centre of \(\Gamma'\). Three conics \(\Gamma_1, \Gamma_2, \Gamma_3\) have a common focus and are such that \(\Gamma_2\) and \(\Gamma_3\) touch at \(P_1\), \(\Gamma_3\) and \(\Gamma_1\) touch at \(P_2\), and \(\Gamma_1\) and \(\Gamma_2\) touch at \(P_3\). Prove that the tangents at \(P_1, P_2, P_3\) meet the corresponding directrices of \(\Gamma_1, \Gamma_2, \Gamma_3\) respectively in three collinear points.
Explain what is meant by an involution of pairs of points on a line. A line \(p\) meets the sides \(BC, CA, AB\) of a triangle \(ABC\) in \(L, M, N\) respectively. If \((L, L'), (M, M'), (N, N')\) are pairs of an involution on \(p\), prove that the lines \(AL', BM', CN'\) are concurrent.
Prove that, if \(A, B, C, D\) are points on a conic \(S\) and \(A, B\) separate \(C, D\) harmonically, \(AB\) is conjugate to \(CD\) with respect to \(S\). \(A, B, C, A', B', C'\) are six points on \(S\) such that \[ \{A, A'; B, C\} = \{B, B'; C, A\} = \{C, C'; A, B\} = -1. \] Prove that \((A, A'), (B, B')\) and \((C, C')\) are pairs in an involution on \(S\).
A point on the conic \(S=y^2-zx=0\) is said to have parameter \(\theta\) if its coordinates are \((\theta^2:\theta:1)\). Find the condition that the line \(lx+my+nz=0\) should cut \(S\) in points which are conjugate with respect to the conic \(\Gamma=x^2+z^2+2yz+2xy=0\). Prove that the polar of the point with parameter \(\theta\) with respect to \(\Gamma\) satisfies this condition, and deduce that there is an infinite number of triangles inscribed in \(S\) and self-conjugate with respect to \(\Gamma\).
\(S\) and \(S'\) are two conics in a plane and \(P\) is a point in the plane. Prove that in general there is a unique point \(P^*\) which is conjugate to \(P\) with respect to both conics. If \(P\) moves on a straight line \(l\), prove that the locus of \(P^*\) is in general a conic \(\Gamma\). When \(S\) and \(S'\) are circles, determine the directions of the asymptotes of \(\Gamma\), and characterize the lines \(l\) for which \(\Gamma\) is a parabola.