Express the radius \(R\) of the circumcircle of a triangle \(ABC\) in terms of the sides, and prove that if \(K\) is its centre and \(O\) is the orthocentre \[ OK^2 = R^2(1-8\cos A\cos B\cos C). \] Prove that if \(A>B>C\), and \(p,q,r\) are the lengths of the perpendiculars from \(A, B, C\) on \(OK\), then \(p-q+r=0\).
A mound on a level plane has the form of a portion of a sphere. At the bottom its surface has a slope \(\alpha\) and at a point distant \(a\) from the bottom the elevation of the highest visible point is \(\beta\). Shew that the height of the mound is \[ a\sin\beta\sin^2\frac{\alpha}{2}\operatorname{cosec}^2\frac{\alpha-\beta}{2}. \]
Give definitions of \(e^z, \sin z, \cos z\) where \(z\) is a complex number and verify that \[ \sin(z_1+z_2) = \sin z_1 \cos z_2 + \cos z_1 \sin z_2. \] Prove that \[ x+\frac{x^4}{4!}+\frac{x^7}{7!}+\dots = \frac{1}{3}e^x+\frac{2}{3}e^{-\frac{x}{2}}\sin\left(\frac{x\sqrt{3}}{2}-\frac{\pi}{6}\right), \] where \(x\) is real.
The distances of a point from the vertices of an equilateral triangle of unknown size are given. Show how the triangle may be constructed by making first a triangle with the lengths of its sides equal to the three given distances.
A variable triangle \(PQR\) inscribed in a circle has the side \(PQ\) parallel to a fixed chord, and \(QR\) passes through the middle point of the chord. Shew that the side \(RP\) also passes through a fixed point.
An aeroplane has an engine-speed equal to that of the wind in which it is flying, and heads continually for a fixed point at its own level. Shew that it moves along a parabola.
Find the locus of the point of intersection of a variable line through a focus of a conic, and a tangent cutting it at a right angle; and shew that it is a circle touching the conic twice. \par Shew that the same is true if the angle is constant but not a right angle.
Two circles in different planes both touch the line of intersection of the planes at the same point. Shew that if a variable plane touches both the circles, it passes through a fixed point \(O\); and that if \(P\) and \(Q\) are the contact points, the product \(OP.OQ\) is constant.
Shew that for a variable normal to a conic the locus of the middle point of the intercept between the axes is a similar and coaxal conic; and shew that two conics may be mutually related in this way.
A conic has eccentricity \(e\) and focus \((a,b)\); and the corresponding directrix is \(lx+my+n=0\). Write down the equation of the conic, and convert it into a form which exhibits the other focus and directrix.