\(ABC\) is a triangle. Points \(D\), \(E\), \(F\) are chosen on \(BC\), \(CA\), \(AB\) such that \(AD\), \(BE\), \(CF\) are concurrent. Points \(P\), \(Q\), \(R\) are chosen on \(EF\), \(FD\), \(DE\) such that \(DP\), \(EQ\), \(FR\) are concurrent. Prove that \(AP\), \(BQ\), \(CR\) are concurrent.
\(ABCD\) is a plane quadrangle, \(AB\) meets \(CD\) in \(E\), \(AC\) meets \(BD\) in \(F\) and \(AD\) meets \(BC\) in \(G\); \(FG\) meets \(AB\), \(CD\) in \(P\), \(P'\), \(GE\) meets \(AC\), \(BD\) in \(Q\), \(Q'\) and \(EF\) meets \(AD\), \(BC\) in \(R\), \(R'\). Prove that \(P\), \(P'\); \(Q\), \(Q'\); \(R\), \(R'\) are pairs of opposite vertices of a complete quadrilateral. What is the dual of this result?
Show that a conic can be represented parametrically, in homogeneous coordinates, by the form \(x:y:z = \theta^2:\theta:1\), by a suitable choice of coordinate system. In the form \(x:y:z = \theta^2:\theta:1\), \(P\) is a fixed point in the plane of a conic \(S\), not lying on \(S\), and \(P\), \(Q\) are points on \(S\) such that \(OP\) and \(OQ\) are conjugate with respect to \(S\). Find the envelope of \(PQ\).
\(A\), \(B\), \(C\), \(D\) are four points on a conic. The tangents at \(A\), \(B\), \(C\), \(D\) meet \(BC'\), \(CD'\), \(DA'\), \(AB\) in \(A_2\), \(B_2\), \(C_2\), \(D_2\). Prove that
Prove that there exists a sphere touching the six edges of the tetrahedron \(ABCD\) internally if, and only if, \(AB + CD = AC + BD = AD + BC\). If, in addition, there exists a sphere touching \(CD\), \(DB\), \(BC\) internally and also touching \(AB\), \(AC\), \(AD\) produced, prove that \(AB = AC = AD\), and that the triangle \(BCD\) is equilateral.
A convex solid bounded by triangular faces is such that, at each vertex, either three or four edges meet. If \(x\) and \(y\) are the numbers of vertices of each type, prove that \[ 3x + 2y = 12. \] Show that, for each solution of this equation in non-negative integers, a polyhedron with the corresponding property actually exists.
Prove that if \(A\), \(B\), and \(C\) are three collinear points and \(P\) is a point not on the same straight line, then the centres of the circles \(PAB\), \(PBC\), and \(PCA\) are concyclic with \(P\).
Prove that the inverse of a circle with respect to a coplanar circle is either a circle or a straight line. Show that the inverse of a system of coaxal circles is in general a similar system, and justify the inverses of the line of centres and the radical axis.
Prove that through four coplanar points there can in general be drawn two parabolas with one rectangular hyperbola. Explain what happens when the four points are the intersections of two rectangular hyperbolas, and relate this case to a theorem concerning the circumcentre of a triangle inscribed in a rectangular hyperbola.
Prove that if the normal at the point \(P(x, y)\) of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) cuts the major and minor axes at \(R\) and \(S\) respectively, and the ellipse again in \(T\), then the ratio of \(PR\) and \(PS\) is a constant (to be found). Find also the length of the chord \(PT\).