\(ABCD\) is a trapezium, with \(AB\) parallel to \(DC\). Lines \(BL\), \(DM\) are drawn to \(AC\), meeting the respective sides \(CD\), \(AB\) of the trapezium, produced if necessary, in \(L\) and \(M\). Prove that the centroid of the trapezium lies on \(LM\).
\(P\) and \(Q\) are two points on a semicircle whose diameter is \(AB\); \(AP\) and \(BQ\) meet in \(N\). Prove that the circle on \(MN\) as diameter cuts the semicircle orthogonally and that \(MN\) is perpendicular to \(AB\).
\(PP'\) is a focal chord of a parabola. Prove that the circle on \(PP'\) as diameter touches the directrix. If the normals to the parabola at \(P\), \(P'\) meet the curve again in \(Q\), \(Q'\) prove that \(PP'\) and \(QQ'\) are parallel.
Two parallel tangents of an ellipse, whose points of contact are \(P\) and \(P'\), are met by a third tangent in \(Q\) and \(Q'\). Prove that \(PQ \cdot P'Q'\) is equal to the square on the semidiameter conjugate to \(PP'\).
Prove that the locus of points from which the two tangents to the conic $$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$$ are perpendicular is the circle (the director circle of the conic) $$(ab - h^2)(x^2 + y^2) - 2(hf - bg)x - 2(gh - af)y + (bc + ca - f^2 - g^2) = 0.$$ If \(P\) is a point of a conic, \(Q\) the centre of curvature of the conic at \(P\), and \(R\) the image of \(Q\) in \(P\), prove that \(P\) and \(R\) are conjugate with respect to the director circle.
A triangle \(PQR\) is such that its vertices lie on the sides \(BC\), \(CA\), \(AB\), respectively, of a fixed triangle. Its sides \(PR\) and \(PQ\) pass through two fixed points \(M\), \(N\) on a fixed line through \(A\). Prove that \(QR\) passes through a fixed point \(L\), and identify this point precisely. State the dual theorem.
In a homography \(T\) on a straight line \(l\), to points \(A\), \(B\) there correspond respectively \(A'\), \(B'\), and \(M\) is a self-corresponding point. If \(M\) are any two points on a straight line through \(M\), \(BA\), \(B'A'\) meet in \(A'\) and \(BB\), \(B'B'\) meet in \(B'\). If \(A'B'\) meets \(l\) in a point \(N\) distinct from \(M\), prove that \(N\) is also a self-corresponding point of \(T\), and deduce that the cross-ratio \((M \text{ } NPP')\), where \(P'\) corresponds to \(P\), is constant for all positions of \(P\) on \(l\). What can be said about \(T\) if \(A'B'\) passes through \(M\)?
\(A\), \(B\), \(C\), \(D\), \(E\) and \(P\) are six points in general position in a plane. Describe and justify straight line constructions for finding the other intersections \(A_1\) and \(B_1\) of the lines \(PA\), \(PB\) with the conic \(S\) through \(A\), \(B\), \(C\), \(D\), \(E\), and deduce a construction for the polar line of \(P\) with respect to \(S\). Give also without proof a construction for the tangent to \(S\) at \(A\).
From a point \(O\) perpendiculars \(OA'\), \(OB'\), \(OC'\), \(OD'\) are drawn to the faces of a tetrahedron \(ABCD\). Prove that pairs of lines such as \((AB, C'D')\), \((BC, A'D')\) are mutually perpendicular. Hence prove that any pair of perpendiculars from \(A\), \(B\), \(C\), \(D\) to the corresponding faces of the tetrahedron \(A'B'C'D'\) are coplanar, and deduce that all these perpendiculars are concurrent.
A circle of radius \(r\) rolls completely round the outside of a closed convex curve \(\mathscr{C}\) of length \(2\pi r\). Show (i) that the centre of the circle traces out a curve \(\mathscr{D}\) of length \(4\pi r\), and (ii) that the region inside \(\mathscr{D}\) but outside \(\mathscr{C}\) has area \(3\pi r^2\).