A triangle has two vertices \(P, Q\) at the ends of a variable diameter of a fixed circle centre \(A\) and the third vertex at a fixed point \(O\) in the plane of the circle. Find the locus of the foot of the perpendicular from \(O\) on to \(PQ\), and deduce, or otherwise prove, that the locus of the orthocentre of the triangle \(OPQ\) is the polar of \(O\) with respect to the circle.
Two points \(P, P'\) on a straight line are related by the equation \[ axx'+bx+cx'+d=0, \] where for a fixed point \(O\) of the line, \(OP=x, OP'=x'\). Prove that the cross-ratio \((PQRS)\) of four points of the line is equal to the cross-ratio \((P'Q'R'S')\) of the four related points. [It may be assumed that \(ad \neq bc\).] Prove that if \(I\) is the point which is related to the point at infinity, and \(J'\) is the point to which the point at infinity is related, then \(IP.J'P'\) is a constant for any point \(P\) and its related point \(P'\).
Prove that through any point in the plane of a set of coaxal circles one and only one circle can be drawn to cut all the circles of the set orthogonally. Hence, or otherwise, show that polars of a point with respect to the circles of a coaxal set are concurrent. Prove also that the ratio of the lengths of the tangents drawn from a point on a fixed circle of a coaxal set to two other fixed circles of the same set is constant.
\(ABCD\) is a tetrahedron, and \(H_1\) and \(H_2\) are the orthocentres of the triangles \(BCD, CAD\) respectively. Prove that, if the perpendicular through \(H_1\) to the plane of \(BCD\) meets the perpendicular through \(H_2\) to the plane of \(CAD\), then the edges \(AB, CD\) are perpendicular. Prove that if the perpendicular through \(H_1\) to the plane of \(BCD\) passes through \(A\), then it must meet the perpendicular through \(H_2\) to the plane of \(CAD\), and also each pair of opposite edges of the tetrahedron are perpendicular.
Prove that a necessary and sufficient condition for concurrence of the normals to a parabola at the three vertices of an inscribed triangle is that the feet of the perpendiculars from the vertex of the parabola on to the sides of the triangle are collinear.
The circumcircle of a triangle \(ABC\) inscribed in a rectangular hyperbola meets the curve again in \(P\). Show that the locus of the feet of the perpendiculars from \(P\) on to the sides of the triangle is a straight line passing through the centre of the hyperbola.
Prove the following results for the number of conics (real and imaginary) which can be drawn through \(n\) distinct given points in general position and to touch \(5-n\) distinct straight lines in general position in a plane. For \(n=5\) or \(0\), one. For \(n=4\) or \(1\), two. For \(n=3\) or \(2\), four. Explain why the results fall together in pairs in this way.
The equation \(ax^2+2hxy+by^2=1\) represents a conic in rectangular Cartesian coordinates. Find the equation of the pair of straight lines through the origin which cut the conic in the four points where it meets the circle of radius \(r\) and centre the origin, and deduce that the semi-lengths of the principal axes of the conic are \(r_1, r_2\), where \(r_1^2, r_2^2\) are the roots of the equation \[ \frac{1}{r^4} - \frac{1}{r^2}(a+b)+ab-h^2=0. \] Find the equations of the principal axes of the conic \[ x^2+6xy-7y^2=8. \]
(i) Prove that \[ \sum_{r=1}^{r=n} \cos^r\theta \sin r\theta = \cot\theta(1-\cos^n\theta \cos n\theta). \] (ii) Without using tables show that \[ \cos \frac{\pi}{10} = \frac{1}{4}\sqrt{(10+2\sqrt{5})}. \]
Establish the existence of the Nine Points Circle of a triangle \(ABC\), and determine the position of the centre \(N\) of this circle in relation to the centroid \(G\), the orthocentre \(H\), and the circumcentre \(O\), of the triangle. If \(A', B', C', U, V, W\) are the mid-points of \(BC, CA, AB, AH, BH, CH\) respectively, and if \(D, E, F\) are the feet of the perpendiculars from \(H\) on to the sides \(BC, CA, AB\) respectively, find for the three triangles \(A'B'C'\), \(UVW\), \(DEF\) as many as possible of the appropriate central points as indicated above for \(ABC\). What is the relation of \(H\) to the triangle \(DEF\)?