Prove that, if the middle points of the coplanar lines \(AB, BC, CD, DA\) are concyclic, \(AC\) is at right angles to \(BD\): deduce that, if the middle points of five of the joins of four points are on a circle, so also is the middle point of the sixth join.
Prove that in successive inversion with regard to two orthogonal circles the order of inversion is immaterial: shew also that, if \(P\) be a point, \(P_1\) and \(P_2\) its inverse points with regard to any two circles not orthogonal, and \(P_{12}, P_{21}\) their inverses, the five points are on a circle cutting the two circles orthogonally.
The lines \(AP, BP\) through fixed points \(A\) and \(B\) are such that the angles made with the line from \(A\) to \(B\) have a constant sum; shew that the locus of \(P\) is a rectangular hyperbola of which \(AB\) is a diameter. Deduce that the points of contact of tangents in a given direction to confocal conics lie on a rectangular hyperbola through the foci.
Two fixed lines which do not intersect are taken in space: shew that in a definite direction one and only one line can in general be drawn to intersect both lines. What are the exceptional directions? Prove also that, if directions be chosen parallel to a fixed plane, the corresponding lines have intercepts between the two fixed lines such that their middle points lie on a line.
Prove that the polar reciprocal of a circle is a conic of which the origin of reciprocation is a focus. Prove that two ellipses with one common focus cannot intersect in four real points.
Find the equation of the normal, the coordinates of the centre of curvature and the equation of the circle of curvature at the point \((am^2, 2am)\) on the parabola \(y^2-4ax=0\). Shew that this circle of curvature surrounds the circle of curvature at the vertex and their radical axis touches the parabola \(2y^2=9ax\).
Prove that there are two points on a quadrant of an ellipse such that the normals are at the same given distance from the centre and that the distance must be less than \(a-b\); shew also that if \(\theta_1, \theta_2\) be the excentric angles of the feet of such a pair of normals, \(a\tan\theta_1\tan\theta_2=b\), where \(2a, 2b\) are the lengths of the axes.
Prove that the ellipse \[ b^2x^2+a^2y^2=a^2b^2, \quad b^2 = a^2(1-e^2) \] is touched at two points by each of the circles \[ x^2+y^2-2\lambda aex + \lambda^2a^2 = b^2, \] \[ x^2+y^2-2\mu aey - \mu^2b^2 = a^2, \] and that, if one point of contact be common to the two circles, \[ a^2\lambda^2+b^2\mu^2 = a^2-b^2. \]
Prove that the equation of any conic inscribed in the rectangle \[ x = \pm a, \quad y = \pm b \] is of the form \[ b^2x^2+a^2y^2-a^2b^2-2\lambda xy + \lambda^2=0: \] and that the two conics of the system, defined by \(\lambda_1\) and \(\lambda_2\), have their eight points of contact with the sides of the rectangle on the conic \[ b^2x^2+a^2y^2-a^2b^2 - (\lambda_1+\lambda_2)xy+\lambda_1\lambda_2=0. \]
A family of conics is such that two given points are the respective poles of two given lines with regard to each conic: shew that the conics have double contact, touching two definite lines at the same points, and that this relation passes into four-point contact if the line joining the two given poles passes through the intersection of the polars.