If \(\theta\) and \(\phi\) are unequal angles less than \(2\pi\), eliminate \(\theta\) and \(\phi\) from the equations \[ x\cos\theta+y\sin\theta = x\cos\phi+y\sin\phi = 2a, \] and \[ 2\cos\frac{\theta}{2}\cos\frac{\phi}{2}=1. \]
(i) If \(O\) is the circumcentre of the triangle \(ABC\), and if \(AO\) meets \(BC\) in \(D\), prove that \[ \frac{AO}{OD} = \frac{\cos(B-C)}{\cos A}. \] (ii) If \(P\) is the orthocentre of the triangle \(ABC\), and if \(OP\) is equally inclined to \(AB\) and \(AC\), prove that the angle \(A\) is \(60^\circ\).
The centre of a fixed circle of radius \(\frac{3}{2}r\) is on the circumference of another fixed circle of radius \(r\). Inside the smaller crescent-shaped area intercepted between the circles is placed a movable circle of radius \(\frac{1}{2}r\). If this circle remains in contact with the circle of radius \(r\), prove that the length of arc described by its centre in moving from one extreme position to the other is \(\frac{13}{12}\pi r\).
If \(x\) is an acute angle and if \(y=\log\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\), prove that \(\cos x \cosh y=1\), and that \[ y = \sin x - \frac{1}{3}\sin 3x + \frac{1}{5}\sin 5x - \dots. \]
If \(x\) and \(\theta\) are real, and \(n\) is a positive integer, express \(x^{2n}-2x^n\cos n\theta+1\) as the product of \(n\) real quadratic factors.
Show that if the lines joining the points \(X, Y\) on the respective sides \(AB, AC\) to the opposite corners of the triangle meet on the median through \(A\), \(XY\) is parallel to \(BC\).
Prove that a line through a point is divided harmonically by the point, the polar of the point with respect to a circle, and the circle itself. \(ABC\) is a triangle and \(D\) is the foot of the perpendicular from \(A\) on \(BC\). The circle on \(AD\) as diameter cuts \(AC, AB\) in \(P, Q\). Prove that the tangents at \(P, Q\) meet on the line joining \(A\) to the middle point of \(BC\).
Reciprocate with respect to a circle the theorem: From a point \(A\) on a circle tangents are drawn to an inner concentric circle and are cut by a third tangent in \(B, C\); then the angle subtended by \(BC\) at the common centre is constant.
Prove that if the six sides of two triangles touch a conic, the six vertices lie on another conic. Prove that if the sides of a triangle touch a parabola, there is a rectangular hyperbola passing through its angular points of which one asymptote is the tangent at the vertex of the parabola.
The centre of a rectangular hyperbola \(S\) is also a focus of another conic \(S'\). A pair of conjugate diameters of \(S\) meet \(S'\) in \(AA', BB'\). Show that the poles with regard to \(S'\) of \(AB, A'B, AB', A'B'\) all lie on the asymptotes of \(S\).