The straight lines joining the vertices \(X\), \(Y\), and \(Z\) of a triangle to a coplanar point \(P\) meet the sides \(YZ\), \(ZX\), and \(XY\), respectively, in \(L\), \(M\), and \(N\). \(L'\) is the harmonic conjugate of \(L\) with respect to \(Y\) and \(Z\), and \(M'\) and \(N'\) are similarly defined. Prove that \(L'\), \(M'\), and \(N'\) are collinear. Prove also that if \(P\) lies on a fixed conic through \(X\), \(Y\), and \(Z\), the straight line \(L'M'N'\) passes through a fixed point.
Two coplanar circles \(C_1\) and \(C_2\) with centres \(O_1\) and \(O_2\) and radii \(a_1\) and \(a_2\) are such that \(C_2\) lies wholly inside \(C_1\). Find the distance \(2h\) between the two limiting points \(L_1\) and \(L_2\) of the system of coaxial circles of which \(C_1\) and \(C_2\) are members in terms of \(a_1\) and \(a_2\) and \(d\) the distance \(O_1O_2\) and show that \[OO_1 = \frac{a_1^2 - a_2^2 + d^2}{2d},\] where \(O\) is the midpoint of \(L_1L_2\).
Prove that if two pairs of opposite vertices of a plane quadrilateral are conjugate with respect to a conic \(S\), then the third pair of vertices are also conjugate. Consider the particular case when the conic is a circle and the quadrilateral is a rectangle.
Prove that the circumcircles of the four triangles formed by four coplanar straight lines have a common point \(P\). Prove also that the orthocentres of the triangles are collinear. State the theorem obtained by inversion with respect to a coplanar circle, and consider the special case when \(P\) is the centre of this circle.
Two conics \(\Sigma\) and \(\Sigma'\) are inscribed in a triangle \(ABC\). A variable tangent to \(\Sigma\) meets \(\Sigma'\) in \(P\) and \(Q\). Prove that if the conic \(S\) through \(A\), \(B\), \(C\), \(P\), and \(Q\) meets \(\Sigma\) in \(R\) and \(S\), the chord \(RS\) passes through a fixed point.
Prove that in general the perpendicular from a vertex on to the opposite face of a tetrahedron is intersected by the straight lines drawn through the orthocentre of, and perpendicular to, the other three triangular faces. Explain what happens if (i) one pair of opposite edges are perpendicular, and (ii) two pairs of opposite edges are perpendicular. In the latter case prove that the centroid of the tetrahedron is the midpoint of the join of the centre of the circumscribing sphere and the meet of the perpendiculars from the vertices on to the opposite faces.
Prove that if two conics \(S\) and \(\Sigma\) are such that a quadrilateral can be inscribed about \(\Sigma\), then if \(P\) is any point of \(S\) and \(Q_1\) and \(Q_2\) are points where two tangents from \(P\) to \(\Sigma\) meet \(S\) again, then the intersection of the other two tangents from \(Q_1\) and \(Q_2\) lies on \(S\).
Explain what is meant by saying that two points \(P\) and \(P'\) of a conic are homographically related. By using the equation \(y^2 = zx\) in homogeneous coordinates, or otherwise, and taking points \((1, 0, 0)\) and \((0, 0, 1)\) as the self-corresponding points of the homography, show that the join \(PP'\) touches a second conic. State the relationship between the two conics.
The sides \(AB\), \(BC\), \(CD\), \(DA\) of a plane quadrilateral are of lengths \(a\), \(b\), \(c\), \(d\), respectively, and the lengths of the two diagonals \(AC\) and \(BD\) are \(x\) and \(y\), respectively. Prove that the area of the quadrilateral is \[\tfrac{1}{4}\{4x^2y^2 - (b^2 + d^2 - a^2 - c^2)^2\}^{\frac{1}{2}}.\] State the particular form of the result when a circle can be inscribed within the quadrilateral.
(i) Solve the equation \[2\cos 5\theta + 10\cos 3\theta + 20\cos \theta - 1 = 0.\] (ii) Prove that if \(\theta_1\), \(\theta_2\), and \(\theta_3\) are values of \(\theta\) which satisfy the equation \[\tan(\theta + \alpha) = \kappa \tan 2\theta\] and are such that no two of them differ by an integral multiple of \(\pi\), then \(\theta_1 + \theta_2 + \theta_3\) is an integral multiple of \(\pi\).