Prove that the common chords of three (intersecting) circles taken in pairs are concurrent. \(D, E, F\) are the feet of the perpendiculars from the vertices \(A, B, C\) to the opposite sides of a triangle whose orthocentre is \(H\). The line \(EF\) meets \(BC\) in \(Q\), and \(P\) is the middle point of \(BC\). Prove that, if \(X\) is the opposite end of the diameter through \(A\) of the circle \(ABC\), then \(XPH\) is a straight line, and, by applying the above theorem to the circles \(ABXC, BCEF, AEHF\), or otherwise, prove that \(XPH\) is perpendicular to \(AQ\), so that \(H\) is also the orthocentre of the triangle \(APQ\).
Prove that the Simson's line of a point \(P\) on the circumcircle of a triangle \(ABC\), with respect to that triangle, is parallel to \(AQ\), where \(Q\) is the point on the circumcircle such that \(PQ\) is perpendicular to \(BC\). The perpendiculars from the vertices of an acute-angled triangle \(ABC\) on the opposite sides meet the circumcircle again in points \(A', B', C'\). Prove that the Simson's lines of any point \(P\) on the circumcircle with respect to the triangles \(ABC\) and \(A'B'C'\) are perpendicular.
The foci of an ellipse are \(S\) and \(S'\). Prove that the tangent and the normal at a point \(P\) of the ellipse are the external and the internal bisectors of the angle \(SPS'\). Prove also that, if \(A\) and \(B\) are two given points and \(l\) a given line, then there is one conic which touches \(l\) and has \(A\) and \(B\) as foci. Find the locus of the point of contact of the conic with the line \(l\), if \(l\) varies so as to pass through a fixed point \(C\) on the line \(AB\) produced.
Two triangles \(ABC, A'B'C'\) in different planes are so related that \(AA', BB', CC'\) meet in a point \(O\). Prove that the lines \(BC, B'C'\) meet in a point \(U\), the lines \(CA, C'A'\) meet in a point \(V\), the lines \(AB, A'B'\) meet in a point \(W\), and that \(U, V, W\) are collinear. A variable plane through \(U, V, W\) meets \(OA, OB, OC\) in \(A'', B'', C''\) respectively. Prove that the vertices of the triangle formed by the lines \(A''U, B''V, C''W\) lie on three fixed lines through \(O\).
\(OA, OB\) are two given lines and \(P\) is a given point in their plane, not lying on either of them. A variable line through \(P\) cuts \(OA, OB\) in \(M\) and \(N\). Prove that the locus of the circumcentre of the triangle \(OMN\) is a hyperbola whose asymptotes are perpendicular to \(OA\) and \(OB\).
\(P\) is a variable point on a given ellipse \(S\) whose equation is \(b^2x^2+a^2y^2=a^2b^2\), and \(L(ak, 0)\), \(M(-ak, 0)\) are two real points on the major axis. The lines \(PL, PM\) cut the ellipse again at \(Q, R\). Prove that \(QR\) touches the ellipse \(\Sigma\) whose equation is \[ \frac{x^2}{a^2} + \frac{(1-k^2)^2}{(1+k^2)^2} \frac{y^2}{b^2} = 1. \] The points \(L, M\) are to be chosen so that \(\Sigma\) touches the line whose equation is \(lx+my=1\). By noting that \(0 < (1-k^2)^2/(1+k^2)^2 < 1\), or otherwise, prove that this is possible only if the given line cuts the ellipse \(S\) in real points, and that the pair \(L, M\) can then be chosen in two ways, the two points in one case being the inverse of the two points in the other case with respect to the circle \(x^2+y^2=a^2\).
\(A\) is a vertex of a rectangular hyperbola, and \(P\) is a point of the hyperbola on the same branch as \(A\). Prove that, if the circle which passes through the centre \(O\) of the hyperbola and touches the hyperbola at \(P\) meets the hyperbola again in two real points, then the angle \(AOP\) is greater than \(\frac{3}{4}\pi\).
\(P\) is the point \((at^2, 2at)\) of the parabola \(y^2=4ax\). Find the parameter of the point \(N\) at which the normal at \(P\) meets the parabola again, and the parameter of the point \(C\) at which the circle of curvature at \(P\) meets the parabola again. For what positions of \(P\) do these two points coincide? Is there a one, one algebraic correspondence on the parabola between (i) \(P\) and \(N\); (ii) \(N\) and \(C\); (iii) \(C\) and \(P\)?
Define an involution of pairs of points on a straight line. A given line \(l\) lies in the plane of a given hyperbola on which \(A\) is a given point. The line joining a variable point \(P\) on the line \(l\) to \(A\) meets the hyperbola again in \(X\), and \(Y\) is the opposite end of the diameter through \(X\). The line \(AY\) meets \(l\) in \(P'\). Prove that the points \(P, P'\) are in involution on \(l\) and that the double points of the involution are the points where \(l\) is met by the lines through \(A\) parallel to the asymptotes.
Shew how to obtain the homogeneous coordinates of the points of a non-singular conic in the parametric form \((\theta^2, \theta, 1)\). The parameters of the points \(A, B\) of the conic are \(\alpha, \beta\). Obtain the equation of the non-singular conic \(\Sigma\), which touches the line \(y=0\) and which also touches the given conic at \(A\) and \(B\), in the form \[ \{x - (\alpha+\beta)y + \alpha\beta z\}^2 + 4\alpha\beta(y^2-zx)=0. \] The lines joining the point \(Y(0,1,0)\) to \(A\) and \(B\) meet the given conic again in \(A'\) and \(B'\), and a conic \(\Sigma'\) is defined as before with \(A\) and \(B\) replaced by \(A'\) and \(B'\). Prove that \(\Sigma\) and \(\Sigma'\) touch the line \(y=0\) at the same point, and identify the line joining their two other points of intersection.