\(A, A'\) are given points inverse with respect to a given circle \(C\), \(A\) being inside \(C\). \(P,Q\) are the extremities of a variable chord passing through \(A\). Prove that the incentre and one of the excentres of the triangle \(A'PQ\) are fixed points. State and prove the corresponding theorem when \(A\) is outside \(C\).
Two given straight lines intersect in \(A\) and \(P\) is a given point. Establish a ruler and compasses construction for the line passing through \(P\) and cutting the given lines in \(B,C\) such that the area of the triangle \(ABC\) is a minimum.
Find the locus of a point which moves so that its distance from a given point \(A\) and a given plane are equal. Determine necessary and sufficient conditions for it to be possible to describe one and only one real sphere to pass through three given points and to touch a given plane.
If \(P\) and \(Q\) are the extremities of a focal chord of a parabola and \(R\) is any point on the diameter through \(Q\), prove that the length of the focal chord parallel to \(PR\) is \(\frac{PR^2}{PQ}\).
Two chords \(PQ, RS\) of a rectangular hyperbola intersect in \(T\) and \(PQ\) is perpendicular to \(QR\). \(QS\) is normal to the hyperbola at \(Q\). Prove that the line passing through \(T\) and concurrent with \(PS\) and \(QR\) passes through the centre of the hyperbola and bisects both \(PR\) and \(QS\). (Note: The question states "PQ is perpendicular to QR". This implies Q is a right angle, which on a hyperbola is unusual. It might mean perpendicular to RS. Assuming text is correct.)
Show that the pairs of tangents drawn from a given point \(P\) to the family of conics touching four given straight lines belong to the involution determined by joining \(P\) to any two pairs of opposite vertices of the quadrilateral formed by the four lines. Deduce the number of members of the family which pass through \(P\).
State and prove the theorem obtained by taking \(I\) and \(J\) in the following theorem to be the circular points at infinity: \(I\) and \(J\) are given points on a fixed conic and \(O\) is the pole of \(IJ\) with respect to the conic. If variable points \(P,Q\) one on each of two given straight lines in the plane of the conic are such that the cross-ratio \(O(PQIJ)\) is constant, then the locus of the pole of \(PQ\) with respect to the conic is a conic passing through \(I\) and \(J\).
\(O\) is a fixed point in the plane of a given conic \(S\). Prove that chords of \(S\) subtending a right angle at \(O\) in general envelope a conic \(\Sigma\). If \(\Sigma\) is a parabola, what type of conic is \(S\)?
Find in radians, correct to two places of decimals, the solutions of:
If \(I\) is the incentre of a triangle \(ABC\), prove that the circumcentre of the triangle \(BIC\) is collinear with \(A\) and \(I\). If \(R, R_1, R_2, R_3\) are the circumradii of the triangles \(ABC, BIC, CIA, AIB\), prove that \(\frac{R_1 R_2 R_3}{2R^2}\) is the radius of the inscribed circle of \(ABC\).