In a triangle \(ABC\), \(G\) is the centroid and \(A'\) is the mid-point of \(BC\). The circles \(CA'G\), \(BA'G\) cut again in \(H\), and the circles \(GA'G\), \(BA'A\) cut again in \(K\). Prove that \(BC\), \(A'K\), \(A'H\) are the tangents at \(A'\) to the circles \(A'HK\), \(A'BG\), \(A'CG\) respectively.
Find in terms of their eccentric angles a necessary and sufficient condition for four points of an ellipse to be concyclic. Four points \(H\), \(P\), \(Q\), \(R\) on an ellipse are concyclic. The circle which touches the ellipse at \(P\) and passes through \(H\) meets the ellipse again in \(P'\). The points \(Q'\) and \(R'\) are similarly defined. Prove that \(H\), \(P'\), \(Q'\), \(R'\) are concyclic.
The point \(P\) on the parabola \(y^2 = 4ax\) has co-ordinates \((at^2, 2at)\). Find (i) the equation of the normal at \(P\), (ii) the equation of the line through \(P\) perpendicular to \(OP\), where \(O\) is the vertex of the parabola. If \(Q\) is an arbitrary point in the plane, show that there are three points on the parabola at which \(OQ\) subtends a right angle. Show further that the normals at these points meet in a point \(R\), and determine the co-ordinates of \(R\) in terms of the co-ordinates of \(Q\).
Find the equation of the rectangular hyperbola whose vertices are at the points \((5, 4)\), \((-3, -2)\).
\(ABC\) 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.