State the theorems of Ceva and Menelaus and prove one of them together with its converse. Corresponding pairs of sides of the triangles \(ABC, PQR\) intersect in points \(A', B', C'\) (i.e. \(BC, QR\) intersect in \(A'\)). If \(AA', BB', CC'\) are concurrent, shew that a sufficient condition for the concurrence of \(PA', QB', RC'\) is that the intersections of \(BQ, CR\), and of \(BR, CQ\), shall lie in \(B'C'\).
Prove that a circle \(C\) will invert into a circle \(C'\) or a straight line, and that two points inverse with respect to \(C\) will invert into points inverse with respect to \(C'\). What happens when the inverse curve is a straight line? Shew that in general eight circles can be drawn to touch three given coplanar circles having real points of intersection.
Find the condition that the two pairs of straight lines represented by the equations \[ ax^2+2hxy+by^2=0, \quad a'x^2+2h'xy+b'y^2=0 \] shall be harmonic conjugates with respect to each other. Hence deduce the equation of the bisectors of the angles between the first pair. Find the equations of the principal axes of the conic \[ 13x^2+37y^2-32xy-14x-34y-35=0. \]
In any tetrahedron prove that the three joins of midpoints of opposite edges are concurrent, and that the point of concurrence is a point of quadrisection of the join of any vertex to the median point of the opposite face.
Prove that the segment of a tangent to a hyperbola cut off between the asymptotes is bisected at the point of contact. Two concentric hyperbolas have one common asymptote and intersect each other in finite points. Shew that the tangent to one of the hyperbolas at a common point has equal intercepts between the two asymptotes not common to the two hyperbolas and between its points of intersection with the other curve.
Shew that three normals to the parabola \(y^2=4ax\) can be drawn from any given point \((\xi, \eta)\). Prove that the centre of the nine-point circle of the triangle formed by the three points at which these normals are drawn has coordinates \(\left(\frac{3\xi-10a}{4}, -\frac{\eta}{8}\right)\).
Define the eccentric angle of a point on an ellipse, and determine the relation between the eccentric angles of the extremities of two conjugate diameters of the ellipse. Prove that, if \(P\) and \(P'\), \(D\) and \(D'\) are pairs of extremities of two conjugate diameters of an ellipse and if \(Q\) is a point on the ellipse such that \(PP'DQ\) is cyclic, then, if the tangent at \(D\) is inclined to the major axis at an angle \(\alpha\), the normal at \(Q\) is inclined to the same axis at an angle \[ \tan^{-1}\left(\cot\alpha \frac{3a^2-b^2\cot^2\alpha}{a^2-3b^2\cot^2\alpha}\right). \]
Give a brief outline of the process of Reciprocation and its application to the solution of geometrical problems. A variable chord \(PQ\) of a given circle subtends a right angle at some fixed point \(O\) inside the circle. Prove that \(PQ\) touches a fixed ellipse having \(O\) as a focus. Discuss the case for \(O\) outside the circle.
Two circles of respective radii \(R, r\) have their centres distance \(d\) apart. Given that \(R^2-d^2=2Rr\), prove that in any triangle inscribed in the first circle and having two of its sides touching the second circle, the third side will also touch the second circle.
(i) Prove that, for \(0 < \theta < \frac{\pi}{2}\), \[ 1 + \frac{1}{2}\cos\theta\cos 2\theta + \frac{1}{3}\cos^2\theta\cos 3\theta + \frac{1}{4}\cos^3\theta\cos 4\theta + \dots \text{ to } \infty \] \[ = -\frac{1}{2\cos^2\theta}\log_e(1-2\cos^2\theta+\cos^4\theta). \] (ii) By using the expressions \(2\cos\theta = z+\frac{1}{z}\), \(2i\sin\theta = z-\frac{1}{z}\), where \(z=e^{i\theta}\), or otherwise, shew that for any positive even integer \(p\) \[ 2^{2p-1}(\cos^{2p}\theta + \sin^{2p}\theta) = \] \[ 2\cos 2p\theta + 2 \cdot {}^{2p}C_1 \cos 2(p-2)\theta + 2 \cdot {}^{2p}C_2 \cos 2(p-4)\theta + \dots + 2 \cdot {}^{2p}C_{p-1} \cos 2\theta + {}^{2p}C_p. \]