Explain what is meant by the cross-ratio of four points (i) on a straight line, (ii) on a conic. \(A, B, C, D\) are four points of a conic \(\Gamma\). A line \(l\) through \(D\) meets \(BC, CA, AB\) respectively in \(P, Q, R\) and meets \(\Gamma\) again in \(S\). Prove that the cross-ratio \((ABCD)\) on \(\Gamma\) is equal to the cross-ratio \((PQRS)\) on \(l\).
\(H\) and \(K\) are two fixed points in the plane of a conic \(S\). Prove that the locus of a point \(P\) which moves so that \(PH\) and \(PK\) are conjugate with respect to \(S\) is a conic \(\Gamma\) through \(H\) and \(K\). Where does \(\Gamma\) meet \(S\)? The triangles \(ABC\) and \(A'B'C'\) are each self-polar with respect to \(S\). By considering pairs of conjugate lines through \(A\) and \(A'\), or otherwise, prove that \(A, B, C, A', B', C'\) lie on a conic.
Two conics \(S, S'\) are circumscribed to a triangle \(ABC\) and touch each other at \(A\). A line \(l\) through \(A\) meets \(S, S'\) again at \(P, P'\) respectively. Prove that the tangents to \(S\) and \(S'\) at \(P\) and \(P'\) meet in a point \(Q\) of \(BC\). Show further that the harmonic conjugate of \(l\) with respect to \(AB, AC\) meets \(S\) and \(S'\) again in two points, the tangents at which also meet in \(Q\).
A variable line \(lx+my+nz=0\) meets the conic \(S \equiv y^2-zx=0\) in two points \(P, P'\) such that the lines \(YP, YP'\) harmonically separate the fixed pair of lines \(ax^2+cz^2+2gzx=0\) through \(Y\). Show that the line \(PP'\) envelops a conic \(\Gamma\), and find the equation of \(\Gamma\) in tangential coordinates. Investigate conditions under which \(\Gamma\) may degenerate.
Three coplanar triangles \(A_1B_1C_1, A_2B_2C_2\) and \(A_3B_3C_3\) are such that \(B_1C_1, B_2C_2, B_3C_3\) meet at \(L\), \(C_1A_1, C_2A_2, C_3A_3\) meet at \(M\), and \(A_1B_1, A_2B_2, A_3B_3\) meet at \(N\), where \(L, M, N\) are collinear. Show that \(A_2A_3, B_2B_3, C_2C_3\) are concurrent (in a point \(O_1\), say). If \(O_2\) and \(O_3\) are similarly defined, show that \(O_1, O_2\) and \(O_3\) are collinear.
Prove Menelaus' theorem, that if a transversal meets the sides \(BC, CA, AB\) of a triangle \(ABC\) in \(D, E, F\) respectively, then \(AF.BD.CE = -FB.DC.EA\). Show that the tangents to the circumcircle of a triangle at its vertices meet the opposite sides in three collinear points.
Explain what is meant by inversion in geometry, and show that the inverse of a circle is either a circle or a straight line. The inverses with regard to a given circle \(\gamma\) of two points \(P, Q\) are two points \(P', Q'\) respectively, and \(C\) is any point on the circle of inversion. Show that the circumcircles of the triangles \(PCQ\) and \(P'CQ'\) meet on \(\gamma\).
Prove that the tangents drawn to a circle from a given external point are equal. The sides of a skew (non-planar) quadrilateral \(ABCD\) each touch a sphere at the four points \(P, Q, R,\) and \(S\). Prove that the quadrilateral \(PQRS\) is cyclic.
Show that if \(S=0\) and \(S'=0\) represent the cartesian equations of two circles, then \(S+kS'=0\) also represents a circle, and explain its relationship to the first two circles. If the tangents from two given points to a variable circle are of given lengths, prove that the variable circle always passes through two fixed points, and state the positions of these two points.
Prove that the locus, as \(t\) varies, of the point whose rectangular coordinates are given by \[ x=at^2+2bt+c, \quad y=a't^2+2b't+c' \] is a parabola. Find the equation of the tangent at the point \(t\), and show that the tangents at the points \(t_1, t_2\) meet at the point \[ at_1t_2+b(t_1+t_2)+c, \quad a't_1t_2+b'(t_1+t_2)+c'. \] Show that the directrix of the parabola is \[ ax+a'y+b^2+b'^2-ac-a'c'=0. \]