Prove that in general there are two points in the plane of three coplanar circles such that the lengths of tangents drawn from either point to the circles is in a given ratio. What happens if the three tangents are of equal length? Shew that there is one fixed circle orthogonal to any circle which passes through a pair of points such that the ratio of the lengths of tangents drawn from either point to three fixed circles is the same, and that this orthogonal circle is independent of the value of the ratio.
Shew that the inverse of a sphere with respect to a centre of inversion on or inside it is a sphere or a plane. Lines are drawn from a given point \(P\) to cut a sphere in variable points \(Q_1\) and \(Q_2\). Shew that if \(Q_1\) lies on a circle, so does \(Q_2\).
Four straight lines in a plane are drawn so that \(AB, CD\) intersect in \(E\), and \(AD, BC\) intersect in \(F\). Prove that the mid-points of \(AC, DB, EF\) are collinear.
Prove that any straight line is cut in pairs of points in involution by conics passing through four fixed points. Hence or otherwise shew that in general through four given points there pass two parabolas and one rectangular hyperbola.
Find the locus of a point from which perpendicular tangents can be drawn to a given conic, noting any special cases. Hence by projection, or otherwise, shew that the locus of a point from which the tangents drawn to a given conic are harmonically conjugate with respect to the joins of the point to two fixed points is another conic.
Points are taken on a given line and through each the perpendicular to the polar of the point with respect to a given conic is drawn. Shew that these perpendiculars all touch a parabola which also has the given line as a tangent.
Prove that a pair of straight lines equally inclined to the axes of a central conic cuts it in four concyclic points, and that every pair of straight lines through these points is equally inclined to the axes. Hence deduce a construction for the centre of curvature at any point of a given central conic.
Shew that in general four fixed planes having a straight line in common intersect any straight line in four points having a fixed cross-ratio. \(l_1, l_2, l_3\) are three given straight lines. Shew that the cross-ratios of the ranges on these lines made by four common transversals are equal.
\(P\) is any point within a triangle \(ABC\), and at a distance \(d\) from its circumcentre, the circumradius being \(R\). If \(L, M, N\) be the feet of the perpendiculars from \(P\) drawn to the three sides \(BC, CA, AB\), shew that:
Shew that if \(P_1, P_2, \dots, P_{2n}\) be vertices of a regular polygon with an even number of sides \(2n\) and \(O\) be the centre of the circumscribing circle, the product of the perpendicular distances from \(O\) on to \(P_1P_2, P_1P_3, \dots, P_1P_n\) is \((\frac{1}{2}a)^{n-1}\sqrt{n}\), where \(a\) is the radius of the circumscribing circle.