Through the vertices \(A, B, C\) of an acute-angled triangle \(ABC\) straight lines \(VAW, WBU, UCV\) are drawn so that the triangles \(ABC, UVW\) are similar. Prove that the circles \(BCU, CAV, ABW\) are equal, and that they meet in a point which is the orthocentre of the triangle \(ABC\) and the circumcentre of the triangle \(UVW\).
State and prove the theorem of Menelaus for a transversal \(LMN\) of a triangle \(ABC\). \(ABCD\) is a given parallelogram; points \(Q, V\) are taken on \(AD, BC\) respectively so that \(QV\) is parallel to \(AB\), and points \(R, W\) are taken on \(AB, CD\) respectively so that \(RW\) is parallel to \(AD\). Prove that \(QR\) meets \(VW\) on the diagonal \(BD\).
Define a hyperbola, and prove that, if \(A, B\) are two given points, then the locus of a point \(P\) which moves so that \(PA-PB\) is constant is one branch of a hyperbola. Two given non-intersecting circles have centres \(A, B\) and radii \(a, b\) respectively (\(a>b\)). \(P\) is the centre of a circle touching each of the given circles, internally or externally. Find the complete locus of \(P\), distinguishing the different cases.
A tetrahedron \(ABCD\) has the property that a sphere can be drawn to touch each of its six edges. Prove that \[ AD+BC=BD+CA=CD+AB. \] Investigate whether the converse result is true that, if a tetrahedron \(ABCD\) has the property that \[ AD+BC=BD+CA=CD+AB, \] then there exists a sphere touching each of its six edges.
The coordinates of a variable point \(T\) of a certain curve are given in terms of a parameter \(t\) by means of the relations \[ x=at^3, \quad y=at, \] where \(a\) is constant. Prove that, if \(P, Q, R\) are three distinct collinear points of the curve, with parameters \(p, q, r\), then \[ p+q+r=0. \] Prove also the converse result that, if \(p+q+r=0\), then the points are collinear. \(A, B, C\) are three points on the curve. The lines \(BC, CA, AB\) meet the curve again in \(L, M, N\) and the lines joining \(A, B, C\) to the origin meet the curve again in \(U, V, W\). Prove that \(LU, MV, NW\) are concurrent.
Tangents \(PL, PM\) drawn to a parabola from a point \(P\) meet the directrix in \(U, V\) respectively. The second tangent from \(U\) meets \(PM\) in \(R\), and the second tangent from \(V\) meets \(PL\) in \(Q\). Prove that the point of intersection of \(QV, RU\) is the orthocentre of the triangle \(PQR\). Hence, or otherwise, show that the foot of the perpendicular from \(P\) to \(QR\) is the focus of the parabola.
The foci of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] are \(S(ae,0)\), \(S'(-ae,0)\), where \(e\) is the eccentricity, and \(P(x_1, y_1)\) is an arbitrary point of the ellipse. Prove that \[ SP = a-ex_1, \quad S'P = a+ex_1. \] Prove that, if \(p\) is the length of the perpendicular from the centre of the ellipse to the tangent at \(P\), then \[ p = \frac{ab}{\sqrt{(SP \cdot S'P)}}. \]
A straight line meets a hyperbola in \(A, B\) and its asymptotes in \(C, D\). Prove that the segments \(AB\) and \(CD\) have the same middle point. The tangent at a point \(P\) of a hyperbola, whose centre is \(O\), meets the asymptotes in \(U, V\). The normal at \(P\) meets the circle \(OUV\) in \(X, Y\). Prove that \(OX, OY\) are the axes of the hyperbola.
\(A, B, C, D\) are four points on a conic \(S\). The lines \(BC, AD\) meet in \(X\); the lines \(CA, BD\) meet in \(Y\); the lines \(AB, CD\) meet in \(Z\). Prove that the triangle \(XYZ\) (the diagonal triangle of the quadrangle \(ABCD\)) is self-polar with respect to \(S\). A general point \(Y'\) is taken on the line \(YZ\). The lines \(CY', BY'\) meet the conic \(S\) again in \(A', D'\), and the lines \(A'Y, D'Y\) meet \(S\) again in \(C', B'\). Prove that \(XYZ\) is also the diagonal triangle of the quadrangle \(A'B'C'D'\).
The tangents to a conic \(S\) at the points \(Z, X\) meet in \(Y\). Taking \(XYZ\) as triangle of reference, obtain the equation of the conic in the form \(y^2-zx=0\). The straight line \[ \lambda x + \mu y + \nu z = 0 \] meets the conic \(S\) in the points \(U, V\). Prove that the equation of the conic \(T\), through \(X, Y, Z, U, V\), is \[ \nu yz + \mu zx + \lambda xy = 0. \] Prove that, if the straight line \(UV\) passes through the fixed point \((a,b,c)\), then the pole of \(UV\) with respect to the conic \(T\) lies on the conic whose equation is \[ (cx+az)^2 = b(ayz+2bzx+cxy). \]