Define the polar plane of a point with regard to a sphere; and shew that if points are taken on a straight line their polar planes pass through another straight line.
The perpendiculars from the angular points of the triangle \(ABC\) on the opposite sides are produced to meet the circumcircle in \(A', B', C'\) respectively. Prove that the lines drawn from \(O\) the centre of the circumcircle to \(A, B, C\) are perpendicular respectively to the sides of the triangle \(A'B'C'\).
Prove the harmonic properties of a complete quadrilateral. If \(A, P, B\) are three points in a straight line, give a geometrical construction for finding the harmonic conjugate of \(P\) with respect to \(A\) and \(B\).
Prove that a point and its inverse with respect to a circle \(C\) invert into a point and its inverse with respect to the circle which is the inverse of the circle \(C\). A chord \(PQ\) of a given circle whose centre is \(O\) touches another given circle. Prove that as \(PQ\) varies the circle \(PQO\) touches a fixed circle.
Prove that if a transversal cut the sides of a triangle \(ABC\) in \(P, Q, R\) respectively, then \(AQ.BR.CP = AR.BP.CQ\). If the transversal moves so that \(PQ:QR\) is constant, prove that \(BR:CQ\) is also constant.
Prove that the polar reciprocal of a circle with respect to any circle is a conic. Reciprocate a system of coaxal circles with respect to a circle whose centre is one of the limiting points. Prove, by reciprocation, that an ellipse and a confocal hyperbola cut at right angles: and that the tangent at any point \(P\) of the hyperbola makes equal angles with the tangents from \(P\) to the ellipse.
Show that if the sum of the squares on a pair of opposite edges of a tetrahedron is equal to the sum of the squares on a second pair of opposite edges, then the two remaining edges are perpendicular to each other.
Prove that the equations of any two circles may by a proper choice of axes be obtained in the form \[ x^2+y^2+2kx+c=0, \quad x^2+y^2+2k'x+c=0, \] and find their angle of intersection. Show that the locus of the pole of a given straight line with respect to a system of coaxal circles is a hyperbola.
Find the equation of the chord joining two points on the ellipse \(x^2/a^2 + y^2/b^2 = 1\) whose eccentric angles are \(\theta\) and \(\phi\). If \(PSQ, PHR\) are focal chords of the ellipse, where \(S\) and \(H\) are the foci; show that the chord \(QR\) is the line \(\frac{x}{a}\cos\theta + \frac{y}{b}\left(\frac{1+e^2}{1-e^2}\right)\sin\theta + 1 = 0\), where \(\theta\) is the eccentric angle of \(P\).
Find the condition that the line \(lx+my+n=0\) should touch \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Show that the locus of the centre of a conic which has a given focus, touches a given straight line and passes through a given point is a conic.