Prove that if \(A\), \(B\), and \(C\) are three collinear points and \(P\) is a point not on the same straight line, then the centres of the circles \(PAB\), \(PBC\), and \(PCA\) are concyclic with \(P\).
Prove that the inverse of a circle with respect to a coplanar circle is either a circle or a straight line. Show that the inverse of a system of coaxal circles is in general a similar system, and justify the inverses of the line of centres and the radical axis.
Prove that through four coplanar points there can in general be drawn two parabolas with one rectangular hyperbola. Explain what happens when the four points are the intersections of two rectangular hyperbolas, and relate this case to a theorem concerning the circumcentre of a triangle inscribed in a rectangular hyperbola.
Prove that if the normal at the point \(P(x, y)\) of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) cuts the major and minor axes at \(R\) and \(S\) respectively, and the ellipse again in \(T\), then the ratio of \(PR\) and \(PS\) is a constant (to be found). Find also the length of the chord \(PT\).
In a tetrahedron \(ABCD\) the edges \(AD\) and \(BC\) are perpendicular, \(AB = CD\), and \(AC = BD\). Prove that \(AB = AC\).
Show that there exists a circle that intersects a conic in the four points in which it meets its directrices. Show further that a quadrilateral can be drawn with one vertex at any point of the circle and inscribed in the circle and circumscribed about the conic. State any peculiarity of such a quadrilateral.
Prove Pascal's Theorem, that the intersections of opposite sides of a hexagon inscribed in a conic are collinear. Use this theorem to devise a construction to find the second point in which any given straight line through a point \(A\) meets the conic through \(A\) and touching two given straight lines \(OB\) and \(OC\) at \(B\) and \(C\) respectively.
Prove that the polars of a point \(P\) with respect to the conics through four fixed points will meet in another point \(Q\). Prove that as \(P\) moves on a given straight line \(l\), the locus of \(Q\) is a conic.
The area of a triangle is to be determined by the measurement of its sides. If the maximum small percentage error in the measurement of the sides is \(e\), prove that if the triangle is acute angled, the maximum percentage error in the calculated value of the area is approximately \(2e\) per cent. Explain briefly how the percentage error could be calculated when the triangle is obtuse.
Prove that \(\cot \theta - 2 \cot 2\theta = \tan \theta\). Hence or otherwise prove that: \[\frac{1}{2} \tan \frac{\theta}{2} + \frac{1}{2^2} \tan \frac{\theta}{2^2} + \ldots + \frac{1}{2^n} \tan \frac{\theta}{2^n} = \frac{1}{2^n} \cot \frac{\theta}{2^n} - \cot \theta.\] Deduce the result \[\frac{1}{\theta} = \cot \theta + \sum_{r=1}^{\infty} \frac{1}{2^r} \tan \frac{\theta}{2^r}.\]