\(AB\) and \(AC\) are two fixed straight lines, and \(O\) is a fixed point. Two circles are drawn through \(O\) one of which touches \(AB\) and \(AC\) at \(D\) and \(E\) respectively and the other touches them at \(F\) and \(G\) respectively. Prove that the circles \(ODF\) and \(OEG\) touch at \(O\).
\(PCQ\) is a given diameter of an ellipse whose centre is \(C\), and \(D\) is any other point on the ellipse. If the area of the triangle \(PDQ\) is a maximum, then \(CP\) and \(CD\) are conjugate diameters. If the perimeter of the triangle \(PDQ\) is a maximum, then the tangents at \(P\) and \(D\) are at right angles.
Prove that a tetrahedron can be constructed so as to have four equal acute-angled triangles for its faces; shew also that in such a tetrahedron the centre of the circumscribing sphere, the centre of the inscribed sphere, and the centre of inertia all coincide, and that the lines joining the middle points of pairs of opposite edges bisect each other at right angles at this same point.
If any one of the three quantities \(ax+bz+cy\), \(by+cx+az\), \(cz+ay+bx\) vanishes, prove that the sum of the cubes of the other two is equal to \[ (a^3+b^3+c^3-3abc)(x^3+y^3+z^3-3xyz). \]
Sum to infinity the series \[ \frac{1}{6} + \frac{1\cdot4}{6\cdot12} + \frac{1\cdot4\cdot7}{6\cdot12\cdot18} + \dots. \]
Solution: Notice that \begin{align*} && (1+x)^{-1/3} &= 1 +\frac{-\frac13}{1!}x +\frac{(-\frac13)(-\frac43)}{2!}x^2 +\frac{(-\frac13)(-\frac43)(-\frac73)}{3!}x^3 +\cdots \\ \Rightarrow && (1-x)^{-1/3} &= 1 + \frac{1}{3\cdot 1!}x + \frac{1\cdot4}{3^2\cdot 2!} x^2+\frac{1\cdot4\cdot7}{3^3 \cdot 3!}x^3 + \cdots \\ \Rightarrow && \left(1-\frac12\right)^{-1/3} &= 1 + \frac{1}{6}+\frac{1\cdot4}{6^2 \cdot 2!} + \frac{1 \cdot 4 \cdot 7}{6^3 \cdot 3!} + \cdots \\ \end{align*} Therefore our sum is \(\sqrt[3]{2}-1\)
\(ABCD\) is a quadrilateral circumscribing a circle and \(a,b,c,d\) are the lengths of the tangents from \(A, B, C, D\) respectively; prove that the sum of a pair of opposite angles is \(2\theta\), where \[ (a+b)(b+c)(c+d)(d+a)\cos^2\theta = (ac-bd)^2. \]
Prove that, if \(\cos(x+iy) = \tan(\xi+i\eta)\), \[ \cosh 2y - 2\cosh 2y \frac{\sin^2 2\xi+\sinh^2 2\eta}{(\cos 2\xi+\cosh 2\eta)^2} + 2 \frac{\sin^2 2\xi-\sinh^2 2\eta}{(\cos 2\xi+\cosh 2\eta)^2} = 1. \]
A triangle is inscribed in the ellipse \(x^2/a^2+y^2/b^2=1\) and has its centre of gravity at the centre of the ellipse; shew that the locus of its circum-centre is \[ a^4x^2+b^4y^2=\frac{1}{16}(a^2-b^2)^2. \]
On a radius \(OA\) of a circular disc as diameter a circle is described, and the disc enclosed by it is cut out. If the remaining solid rest in a vertical plane on two rough pegs in a horizontal plane subtending an angle \(2\alpha\) at the centre, shew that the greatest angle that \(OA\) can make with the vertical is \(\sin^{-1}(3\sin 2\lambda \sec\alpha)\), where \(\lambda\) is the angle of friction at the pegs.
\(ABCD\) is a rhombus formed by four light rods smoothly jointed at their ends and \(PQ\) is a light rod smoothly jointed at one end to a point \(P\) in \(BC\) and at the other end to a point \(Q\) in \(AD\). Two forces each equal to \(F\) are applied at \(A\) and \(C\) in opposite directions along \(AC\). Prove that the stress in \(PQ\) is \[ F.AB.PQ/AC(AQ\sim BP). \]