Problems

\(D, E, F\) are the feet of the perpendiculars from the vertices on the opposite sides of the triangle \(ABC\). Prove that if \(DE\) meets \(AB\) in \(R\) and \(RQ\), drawn parallel to \(BC\), meets \(AC\) in \(Q\), then \(FQ\) will bisect \(BC\).

Prove that the inverse of a circle is a circle or a straight line and find in each case the position of the point into which the centre of the circle inverts. Two circles \(A\) and \(B\) are given. A variable circle \(C\) passes through the centre of \(A\) and touches \(B\). Prove that the common chord of \(A\) and \(C\) touches a fixed circle coaxal with \(A\) and \(B\).

Prove that the points of intersection of pairs of opposite sides of a hexagon inscribed in a conic are collinear. Prove that if a variable conic through four fixed points \(A, B, C, D\) meets fixed lines through \(A\) and \(B\) in \(X\) and \(Y\), then \(XY\) passes through a fixed point on \(CD\).

Prove that the anharmonic ratio of the points in which a variable tangent cuts four fixed tangents to a conic is constant. Find the anharmonic ratio in which a variable tangent to the parabola \(y^2 = 4ax\) cuts the tangents at the points \((at_r^2, 2at_r)\), where \(r=1, 2, 3, 4\).

Find the condition that the line \(lx+my+n=0\) should be (i) a tangent, (ii) a normal to \(x^2/a^2+y^2/b^2=1\). The normal at the point on this ellipse whose eccentric angle is \(\phi\) meets the ellipse again in \(P\), prove that the tangent to the ellipse at \(P\) is \[ \frac{x}{a}\cos\phi\left(1 - \frac{c^2}{a^2}\cos^2\phi - \frac{c^2}{b^2}\sin^2\phi\right) + \frac{y}{b}\sin\phi\left(1 + \frac{c^2}{a^2}\cos^2\phi + \frac{c^2}{b^2}\sin^2\phi\right) + \frac{c^2}{ab}\left(\frac{\cos^2\phi}{a^2}+\frac{\sin^2\phi}{b^2}\right) = 0, \] where \(c^2=a^2-b^2\).

If \(lx+my+n=0\) is a straight line referred to rectangular axes, interpret the equations:

1. \(al+bm+cn=0\),
2. \((al+bm+cn)(a'l+b'm+c'n)=(a''l+b''m+c''n)^2\),
3. \((al+bm+cn)(a'l+b'm+c'n)=\kappa(l^2+m^2)\).

Find the coordinates of the centre of the conic in (iii).

Find the condition that the line \(lx+my+nz=0\) should touch the conic \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0, \] and, if the coordinates are areals, find the condition that the conic is a parabola. Prove that, the coordinates being areals, the locus of the pole of the line \[ lx+my+nz=0, \] with respect to the system of parabolas circumscribing the triangle of reference, is \[ \{x(my+nz-lx)\}^{1/2} + \{y(nz+lx-my)\}^{1/2} + \{z(lx+my-nz)\}^{1/2} = 0. \]

Find an expression for all angles which have the same sine as \(\alpha\). Find all the solutions of the equation \[ \sin 4\theta - \cos 3\theta = 0. \] Prove that \[ \sin^2 \frac{\pi}{14} + \sin^2 \frac{3\pi}{14} + \sin^2 \frac{5\pi}{14} = \frac{5}{4}. \]

Prove that

1. \(1-\cos^2\alpha - \cos^2\beta - \cos^2\gamma - 2\cos\alpha\cos\beta\cos\gamma\) \[ = -4\cos\frac{1}{2}(\alpha+\beta+\gamma)\cos\frac{1}{2}(-\alpha+\beta+\gamma)\cos\frac{1}{2}(\alpha-\beta+\gamma)\cos\frac{1}{2}(\alpha+\beta-\gamma). \]
2. \(\cos\alpha+\cos\beta+\cos\gamma+\cos(\alpha+\beta+\gamma) = 4\cos\frac{1}{2}(\beta+\gamma)\cos\frac{1}{2}(\gamma+\alpha)\cos\frac{1}{2}(\alpha+\beta)\).

Eliminate \(\theta\) from \begin{align*} a\sin 2\theta &= p\cos\theta+q\sin\theta, \\ b\cos 2\theta &= p\cos\theta-q\sin\theta. \end{align*}

In a triangle prove that, with the usual notation,

1. \(1/r_1 + 1/r_2 + 1/r_3 = 1/r\),
2. \(16R r r_1 r_2 r_3 = a^2b^2c^2\).

The intercepts made by the inscribed circle on the lines drawn from the vertices of a triangle \(ABC\) to the circum-centre are of lengths \(\alpha, \beta, \gamma\). Prove that \[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} = \frac{1}{4r^2} - \frac{1}{8rR \cos A \cos B \cos C}. \]