\(AB\) is a diameter of a circle whose centre is \(O\); \(ODC\) and \(BEC\) are straight lines cutting the circle in \(D\) and \(E\) respectively; \(OB\) is produced to \(F\) so that \(OF=OC\); \(AE\) produced intersects \(CF\) in \(G\). Prove that \(DG\) is a tangent to the circle.
Shew how to determine by a geometrical construction the focus and directrix of a parabola of which two parallel chords are completely given.
Three conics \(A, B, C\) touch two given straight lines. \(P\) is the intersection of the other common tangents of \(B\) and \(C\), \(Q\) that of the other common tangents of \(C\) and \(A\), \(R\) that of the other common tangents of \(A\) and \(B\). Prove that \(P, Q\) and \(R\) are collinear.
Shew that, if the equation of a circle in areal coordinates is in the form \[ \phi(x,y,z) \equiv a^2yz + b^2zx + c^2xy - (lx+my+nz)(x+y+z) = 0, \] the square of the length of a tangent from an external point \((x,y,z)\) is \(-\phi(x,y,z)\). Shew also that this circle will cut the circumscribing circle orthogonally if \[ la\cos A + mb\cos B + nc\cos C = abc. \]
If \begin{align*} a(y^2+z^2-x^2) &= b(z^2+x^2-y^2) = c(x^2+y^2-z^2), \\ \text{and } x(b^2+c^2-a^2) &= y(c^2+a^2-b^2), \end{align*} prove that \[ a^3+b^3+c^3 = (b+c)(c+a)(a+b). \]
If \[ x = \cfrac{1}{a_1 + \cfrac{1}{a_2 + \dots + \cfrac{1}{a_{r-1} + \cfrac{1}{a_r + \cfrac{1}{a_{r+1} + \dots + \cfrac{1}{a_n}}}}}}, \] prove that the continued fraction \[ \cfrac{1}{a_1 + \dots + \cfrac{1}{a_{r-1} + \cfrac{1}{a_{r+1} + \dots + \cfrac{1}{a_n}}}}, \] where the constituent \(a_r\) is omitted, is with the usual notation equal to \[ \frac{x(p_{r-1}q_{r-1} - p_{r-2}q_r) + p_{r-2}p_r - p_{r-1}^2}{x(q_{r-1}^2 - q_{r-2}q_r) + p_r q_{r-2} - p_{r-1}q_{r-1}}. \]
Given \(F\{s^2(z-x), s^3(z-y)\} = 0\), where \(s=x+y+z\), prove that \[ (s-x)\frac{\partial z}{\partial x} + (s-y)\frac{\partial z}{\partial y} = s-z. \]
\(PQ\) is a chord of the ellipse \(x^2/a^2+y^2/b^2=1\) normal at \(P\). Find the maximum and minimum values of \(PQ\), and shew that if \(e>1/\sqrt{2}\), the minimum value of \(PQ\) is \(3\sqrt{3}a^2b^2/(a^2+b^2)^{3/2}\) and discuss the cases when \(e<1/\sqrt{2}\).
If \[ \frac{a \sin^2 x + b \sin^2 y}{b \cos^2 x + c \cos^2 y} = \frac{b \sin^2 x + c \sin^2 y}{c \cos^2 x + a \cos^2 y} = \frac{c \sin^2 x + a \sin^2 y}{a \cos^2 x + b \cos^2 y}, \] shew that \[ a^3+b^3+c^3-3abc = 0. \]
A chain of length 20 feet and weight 10 lbs. is stretched nearly straight between two points at different levels. Assuming that vertically below the middle point of the chord the chain is approximately parallel to the chord and that the tension there is 100 lbs. weight, prove that the sag measured vertically from the middle point of the chord is approximately 3 inches.