A straight line meets the sides \(BC, CA, AB\) of a triangle at \(L, M, N\) respectively. Prove that the circles \(ABC, AMN, BNL, CLM\) have a common point \(O\). An arbitrary line through \(O\) meets the circles \(AMN, BNL, CLM\) again at \(D, E, F\) respectively. Prove that \(AD, BE, CF\) are parallel.
The ridges of two roofs are horizontal and at right angles to each other, and the inclination of each roof to the horizontal is \(\theta\). A vertical chimney casts shadows on them making angles \(\alpha\) and \(\beta\) with their ridges. Show that \[ \cos^2\theta = \cot\alpha \cot\beta. \]
A figure consisting of a circle \(S\) and two points \(P, Q\) inverse with respect to \(S\) is inverted with respect to a coplanar circle \(\Gamma\), whose centre does not lie on \(S\), into a circle \(S'\) and two points \(P', Q'\). Prove that \(P', Q'\) are inverse points with respect to \(S'\), and state the corresponding result when the centre of \(\Gamma\) does lie on \(S\). \(A, B, C, D\) are four coplanar points. \(A'\) is the inverse of \(A\) with respect to the circle \(BCD\), \(B'\) is the inverse of \(B\) with respect to the circle \(CDA\), and \(C'\) is the inverse of \(C\) with respect to the circle \(DAB\). Prove that the circles \(A'BC, AB'C, ABC'\) have a common point.
Find the equation of the tangent to the parabola \(y^2=4ax\) at the point \((at^2, 2at)\). An equilateral triangle is circumscribed to a parabola. Prove that the line joining any vertex of the triangle to the point of contact of the opposite side passes through the focus of the parabola.
Find the condition that the line \(lx+my+n=0\) should be a normal to the ellipse \(x^2/a^2+y^2/b^2=1\). Points \(P, Q\) are taken one on each of two similar and similarly situated ellipses with a common centre at \(O\), and the angle \(POQ\) is bisected by the major axis of the ellipses. Prove that \(PQ\) is always normal to a fixed ellipse.
The perpendicular lines \(l, m\) intersect at \(K\); \(M\) is a fixed point on \(m\) (other than \(K\)) and \(P\) a variable point on \(l\). The points \(Q, R\) are constructed on \(l\) as follows: \(MQ\) is perpendicular to \(MP\) and \(R\) is the inverse point of \(Q\) with respect to the circle through \(P\) with centre at \(K\).
The lines \(a,b,c\) in a plane are concurrent at \(V\); the pairs of points \(A\) and \(A'\), \(B\) and \(B'\), \(C\) and \(C'\) lie on \(a,b,c\) respectively. \(BC\) meets \(B'C'\) in \(L\); \(BC'\) meets \(B'C\) in \(L'\). The pairs of points \(M\) and \(M'\), \(N\) and \(N'\) are defined similarly. Prove that the points \(L, L', M, M', N, N'\) are the vertices of a complete quadrilateral. If the configuration is erased except for the point \(V\) and the points \(L, L', M, M', N, N'\), prove that the lines \(a,b,c\) can be recovered.
The five pairs of points \(A_1, A_2\); \(A_2, A_3\); \(A_3, A_4\); \(A_4, A_5\); \(A_5, A_1\) are all conjugate with respect to a conic \(S\). Prove that the pair of tangents from \(A_1\) to \(S\) separates harmonically both the pair of lines \(A_1A_2, A_1A_5\) and the pair \(A_1A_3, A_1A_4\). Give a construction for this pair of tangents when they are real.
Points \(X, Y, Z\) lie respectively on the sides \(BC, CA, AB\) of the triangle \(ABC\) in such a way that \(AX, BY, CZ\) are concurrent; points \(A', B', C'\) lie on \(YZ, ZX, XY\) in such a way that \(XA', YB', ZC'\) are concurrent. Prove that \(AA', BB', CC'\) are concurrent.
Define a homography on a non-singular conic \(S\). Under a given homography on \(S\) to variable points \(P, Q\) correspond \(P', Q'\); prove that the locus of the meet of \(PQ'\) and \(P'Q\) is a straight line. Calling this line the cross-axis of the homography, prove that, if \(l\) is any line of the plane of \(S\) and \(A, A'\) any pair of distinct points lying on \(S\) and not on \(l\), then there is one and only one homography on \(S\) whose cross-axis is \(l\) and under which \(A'\) corresponds to \(A\). Prove that, if for two homographies there is only one pair \(A, A'\) of points on \(S\) such that under both homographies \(A'\) corresponds to \(A\), then the cross-axes of the two homographies meet on \(AA'\).