On the surface of a sphere, centre \(O\), are four points \(A, B, C, D\). Prove that \(AB\) is perpendicular to \(CD\) if and only if \[ \cos\angle AOC + \cos\angle BOD = \cos\angle AOD + \cos\angle BOC. \]
\(S\) is a circle and \(K\) a point outside it; \(\alpha\) is a given acute angle. Prove that there are precisely two pairs of points, \(Q, R\), such that \(Q\) and \(R\) are inverse points with respect to \(S\), \(KQ=KR\) and \(\angle QKR = 2\alpha\). \(\Sigma_1, \Sigma_2\) and \(\Sigma_3\) are three circles; \(\Sigma_2\) and \(\Sigma_3\) meet in two distinct points, both outside \(\Sigma_1\) and are not orthogonal. Prove that there are two positions for a point \(P_1\) such that its inverses, \(P_3, P_2\), with respect to \(\Sigma_2\) and \(\Sigma_3\) are themselves inverse points with respect to \(\Sigma_1\).
At points \(P, Q, R\) of a parabola tangents are drawn to make a triangle \(LMN\). Prove that the area of \(LMN\) is half the area of \(PQR\), and that the line joining the centroids of the two triangles is parallel to the axis of the parabola.
Prove that, if two rectangular hyperbolas intersect in four points \(A, B, C, D\), then any conic through \(A, B, C, D\) is a rectangular hyperbola. By reciprocating with respect to an arbitrary circle, or otherwise, prove that the director circles of a family of conics touching four fixed lines form a coaxial system.
Define an involution on a line. The six sides of a complete quadrangle cut a general line of the plane in six points; prove that the pairs of points arising from intersections with opposite sides of the quadrangle are pairs in an involution on the line. For what positions of the line does this cease to be true? Given a line and two pairs of points on it, \(P, P'\) and \(Q, Q'\), and a fifth point on it, \(R\), give a construction using only a ruler to find the point \(R'\) on the line which is paired with \(R\) under the involution which pairs \(P\) with \(P'\) and \(Q\) with \(Q'\).
(i) Two conics, \(S_1\) and \(S_2\), have double contact. \(P_1\) is a point which varies on \(S_1\) and the tangent to \(S_1\) at \(P_1\) cuts \(S_2\) at \(Q_2\) and \(R_2\); find the locus of the harmonic conjugate of \(P_1\) with respect to \(Q_2\) and \(R_2\). (ii) Two conics touch at \(A\) and cut at \(B\) and \(C\); their two common tangents, other than that at \(A\), intersect at \(X\). Prove that the tangent at \(A\) and \(AX\) cut \(BC\) at points which are harmonic conjugates with respect to \(B\) and \(C\).
A variable conic, \(S\), passes through the fixed points \(A, B, C\) and touches the fixed line \(l\), which does not contain \(A, B\) or \(C\). Prove that the locus of the pole of \(BC\) with respect to \(S\) is a conic inscribed in the triangle \(ABC\). Determine the points in which this locus meets \(l\).
On a conic, \(S\), are two points, \(A\) and \(B\); \(L\) is a variable point in the plane. \(AL, BL\) cut \(S\) again in \(Q, R\); \(AR, BQ\) meet in \(M\). Prove that, if \(L\) traces out a conic through \(A\) and \(B\), \(M\) traces out a conic through \(A, B\) and the other two points of intersection of \(S\) with the locus of \(L\). If the tangents at \(L\) and \(M\) to their respective loci meet at \(T\), prove the locus of \(T\) to be a straight line.
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 \(P'\).