Prove that there is a point \(P\) in the plane of a triangle \(ABC\) such that the angles \(\angle BCP, \angle CAP, \angle ABP\) are equal. If \(Q\) is the point such that \(\angle CBQ, \angle ACQ, \angle BAQ\) are all equal, prove that \(\angle BCP = \angle QBC\).
Define a coaxal system \(\Sigma\) of circles in a plane. Prove that the circles orthogonal to every circle of \(\Sigma\) form a coaxal system \(\Sigma'\). Prove that, if \(\Sigma_1, \Sigma_2\) are two coaxal systems, they have a circle in common if, and only if, \(\Sigma_1', \Sigma_2'\) have a circle in common. In what circumstances do \(\Sigma\) and \(\Sigma'\) have a circle in common? (Throughout the question the term ``circle'' is taken to include the degenerate circles, i.e. straight lines and point-circles.)
Prove that the lines joining the mid-points of the three pairs of opposite sides of a quadrangle are concurrent. From the mid-point of each side of a cyclic quadrangle is drawn the perpendicular to the opposite side. Prove that the six perpendiculars are concurrent.
A parabola touches each side of a triangle. Prove that its directrix passes through the orthocentre of the triangle.
A conic \(S\) has \(H, h\) as one focus and the corresponding directrix; the chord \(PQ\) cuts \(h\) at \(K\). Prove that \(KH\) is one bisector of the angle \(PHQ\). State the analogous result in which the chord \(PQ\) is replaced by the tangent at \(P\). Two lines \(h, l\) and a point \(H\) are given in a plane. Find a construction for the point of contact with \(l\) of a conic \(S\) which touches \(l\) and for which \(H\) and \(h\) are focus and corresponding directrix.
Explain what is meant by the statement: ``\(P\) corresponds to \(P'\) (or \(P \to P'\)) in a homography on the points of a line.'' Show that in general a homography on a line is uniquely defined by three corresponding pairs of points. \(A, B, C\) are three distinct points of a line and \(T\) is the homography determined by \(A \to B\), \(B \to C\), \(C \to A\). If \(L\) is any other point of the line and \(L \to M\), \(M \to N\) in \(T\), prove that \(N \to L\) in \(T\).
Three points \(A, B, C\) lie in the plane of a conic \(S\). Prove that in general it is possible to find two triangles \(LMN\) inscribed in \(S\) whose sides \(MN, NL, LM\) pass through \(A, B, C\) respectively, but that there exists an infinite number of such triangles if the triangle \(ABC\) is self-polar with respect to \(S\).
The lines joining a point \(P\) to the vertices of a triangle \(ABC\) meet the opposite sides in \(D, E, F\), and the harmonic conjugates of \(P\) with respect to the pairs of points \((A, D), (B, E), (C, F)\) are \(L, M, N\). Prove that the triangle \(LMN\) is circumscribed to the triangle \(ABC\), and that there exists a conic touching \(MN, NL, LM\) at \(A, B, C\) respectively.
The point \(H\) lies on an axis of a confocal system of central conics. Prove that the locus of the points of contact of tangents from \(H\) to the conics of the system is a circle through the two foci on the other axis. Prove also that the locus of the points of contact of tangents parallel to a fixed line is a rectangular hyperbola concentric with the system.
A conic \(S\) is given parametrically in the form \[ x:y:z = t^2:t:1. \] If \(P\) is the point of \(S\) whose parameter is \(t\), find the equation of the conic \(S'\) which touches \(S\) at \(P\), touches the line \(y=0\) at the point \((0, 0, 1)\), and passes through the point \((0, 1, 0)\). Find also the parameter of the remaining intersection \(Q\) of \(S\) and \(S'\), and the equation of the envelope of the line \(PQ\) as \(P\) varies on \(S\).