A point \(D\) is taken on the minor arc \(BC\) of the circumcircle of an equilateral triangle \(ABC\), and \(U\) is the fourth vertex of the parallelogram \(BDCU\) (in which \(DB, DC\) are adjacent sides). Prove that the triangle \(UAD\) is isosceles.
Given a triangle \(ABC\) and a point \(P\) on its circumcircle, it is known that the feet of the perpendiculars from \(P\) to the sides of the triangle lie on a straight line, the Simson or pedal line of \(P\) with respect to the triangle. Prove that, if the line through \(P\) perpendicular to \(BC\) meets the circle again in \(L\), then the pedal line of \(P\) is parallel to \(AL\). Two given circles meet at distinct points \(B, C\); an arbitrary line through \(B\) cuts the first circle at \(A_1\) and the second at \(A_2\), and an arbitrary line through \(C\) cuts the first circle at \(P_1\) and the second at \(P_2\). Prove that the angle between the pedal line of \(P_1\) with respect to the triangle \(A_1BC\) and the pedal line of \(P_2\) with respect to the triangle \(A_2BC\) is independent of the choice of the two arbitrary lines.
A direct common tangent of two non-intersecting circles touches the first at \(P\) and the second at \(Q\); a circle through \(P\) and \(Q\) cuts the first circle again in \(A\) and the second in \(B\); the line \(AB\) cuts the first circle again in \(L\) and the second in \(M\). Prove that \(PA\) is parallel to \(QM\) and that \(QB\) is parallel to \(PL\); and that the parallelogram whose sides lie along these four lines has as its diagonal, other than \(PQ\), the radical axis of the two circles.
A variable point \(P\) is taken on a given ellipse of foci \(A, B\), and \(S\) is the escribed circle opposite \(P\) of the triangle \(PAB\). Prove that the centre of \(S\) lies on the normal at \(P\) to the ellipse and that the length of the tangents from \(P\) to the circle is independent of the position of \(P\) on the ellipse. Prove also that, whatever the eccentricity of the ellipse, the length of these tangents is less than that of the major axis of the ellipse.
A straight line is drawn to cut a hyperbola in \(A, B\) and its asymptotes in \(P, Q\). Prove that the segments \(AB\) and \(PQ\) have the same middle point. Find the locus of the centroid (centre of gravity) of a triangle whose sides are the two asymptotes of a given hyperbola and a variable tangent.
The tangents at two points \(A, B\) of a parabola meet at \(T\) and the normals at \(A, B\) meet at \(N\), and it is given that the line \(TN\) bisects the chord \(AB\). Prove that either \(AB\) is perpendicular to the axis of the parabola or \(AB\) passes through the focus.
The tangents to a conic at two points \(A, B\) meet in \(T\), and an arbitrary line through \(T\) meets the conic in \(C, D\). The lines \(TA, TB\) meet the tangent at \(D\) in \(L, M\), and the lines \(CA, CB\) meet the tangent at \(D\) in \(X, Y\). The line \(AB\) meets the tangent at \(D\) in \(U\). Prove that the points \(U, D\) separate each of the pairs \(L, M\) and \(X, Y\) harmonically. Prove that the conic through \(A, B, L, M, C\) touches the given conic at \(C\).
Prove Pappus's theorem that, if \(A, B, C\) and \(P, Q, R\) are two triads of collinear points on (distinct) coplanar lines, then the three points of intersection \((BR, CQ), (CP, AR), (AQ, BP)\) are collinear. Prove further that, if the triads \(A, B, C\) and \(P, Q, R\) are on two skew lines, and if \(O\) is an arbitrary point of space, then the transversals from \(O\) to the three line-pairs \((BR, CQ), (CP, AR), (AQ, BP)\) are coplanar.
The rectangular hyperbola \(xy=c^2\) meets the ellipse \(b^2x^2+a^2y^2=a^2b^2\) in four real points \(A, B, C, D\). Prove that the area of the parallelogram \(ABCD\) is \(2\sqrt{(a^2b^2-4c^4)}\).
Prove that the equations of two given conics through four distinct points can be expressed in terms of general homogeneous coordinates in the form \begin{align*} ax^2+by^2+cz^2 &= 0, \\ x^2+y^2+z^2 &= 0. \end{align*} Prove that the pole, with respect to any conic through the four points, of the line \(\lambda\) whose equation is \[ Lx+My+Nz=0, \] lies on the conic \(S\) whose equation is \[ (b-c)Lyz+(c-a)Mzx+(a-b)Nxy=0. \] Prove also that, if the line \(\lambda\) varies so as to pass through the point of intersection of the given lines \[ L_1x+M_1y+N_1z=0, \quad L_2x+M_2y+N_2z=0, \] then the conic \(S\) passes through a fixed point, and determine its coordinates.