Four coplanar lines, taken in sets of 3, form 4 triangles; prove that the circumcircles of these 4 triangles meet in a point (P) and that their orthocentres lie on a line (l). Prove also that, if Q is a variable point on the line l, the perpendicular bisector of PQ is cut by the 4 original lines in 4 points whose mutual distances are in constant ratios.
The normals at the points \(P,Q\) of an ellipse are perpendicular and meet the ellipse again in \(P',Q'\) respectively. Prove that, if \(C\) is the centre of the ellipse, the sectorial areas \(CPP', CQQ'\) are equal.
If \(\alpha, \beta\) are the roots of the quadratic \[ ax^2+2hx+b+\kappa(a'x^2+2h'x+b')=0, \] prove that numbers \(p,q\), independent of \(\kappa\), can be found such that \((p-\alpha)(p-\beta)=q^2\), and that \((p\pm q)\) are the roots of the quadratic \[ (ax+h)(h'x+b')-(hx+b)(a'x+h')=0. \] Taking \(x\) as the abscissa of any point, give a geometrical interpretation of the preceding result.
Prove that \[ \frac{nx^{2n-1}}{x^{2n}-1} = \frac{x}{x^2-1} + \sum_{r=1}^{n-1} \frac{x-\cos r\alpha}{x^2-2x\cos r\alpha+1}, \] where \(\alpha=\pi/n\); and deduce that \[ \sum_{r=1}^{n-1} \frac{\cos r\alpha}{\cos r\alpha - \cos\theta} = \frac{n\cos(n-1)\theta}{\sin\theta\sin n\theta}-\csc^2\theta. \]
Two adjacent sides of a parallelogram are of lengths \(a,b\) and include an angle \(\alpha\), and a rhombus is described with one angular point on each of the 4 sides of the parallelogram or on these sides produced. Prove that the least area of the rhombus is \[ ab\sin^2\alpha/(1+\sin\alpha). \]
If the bisectors of the angle \(A\) of the triangle \(ABC\) meet \(BC\) in \(D,D'\), prove that the radius of the circle inscribed in the triangle \(ADD'\) is \[ bc/4R \cos\tfrac{1}{2}(B-C)\{\cos\tfrac{1}{4}(B-C)+\sin\tfrac{1}{4}(B-C)\}, \] where \(R\) is the radius of the circumcircle of the triangle \(ABC\), and \(B>C\).
A variable chord \(PQ\) of a curve passes through a fixed point \(O\) and \(M\) is the middle point of \(PQ\); the normals at \(P,Q\) to the curve meet the line through \(O\) perpendicular to \(OPQ\) in \(P',Q'\). Prove that the middle point of \(P'Q'\) lies on the normal at \(M\) to the locus of \(M\).
Find the locus of centres of curvature of the curve given by the equations \[ x=\cos\theta+\theta\sin\theta, \quad y=\sin\theta-\theta\cos\theta, \] where \(\theta\) is a parameter. Draw a graph of the two curves for \(0\le\theta\le\pi\).
Two coplanar forces of magnitudes \(P,Q\) and inclined at an angle \(\alpha\) act through the fixed points \(A,B\) respectively. Prove that, if each force is rotated through any the same angle in the same sense, their resultant passes through a fixed point at a distance \[ \tfrac{1}{2}AB(P^2+Q^2-2PQ\cos\alpha)^{\frac{1}{2}}/(P^2+Q^2+2PQ\cos\alpha)^{\frac{1}{2}} \] from the middle point of \(AB\).
A thin smooth elliptic tube of axes \(2a, 2b\) (\(a>b\)) is attached by light spokes to a horizontal axis which passes through the centre of the ellipse and is perpendicular to its plane. The weight of the tube is \(W\) and its centre of gravity is on the major axis at a distance \(d\) from the centre; and a particle of weight \(w\) is placed in the tube. Prove that there are 2 or 4 positions of equilibrium according as \(d >\) or \(< (a^2-b^2)w/aW\).