Prove that, if \(BCMN, CANL, ABLM\) are three circles and the lines \(AL, BM, CN\) cut the circle \(LMN\) again in \(P, Q, R\) respectively, the triangle \(PQR\) is similar to \(ABC\).
Prove that the circumcircles of the triangles formed by sets of three out of four given lines meet in a point.
\(O\) is a fixed point and \(l\) is a fixed line in the plane of a conic \(S\). If the foot of the perpendicular from \(O\) to a variable tangent of \(S\) always lies on \(l\), show that \(S\) is a parabola with focus \(O\) having \(l\) as the tangent at its vertex. A rectangular sheet of paper \(ABCD\) is folded so that \(C\) lies on \(AB\). Show that the crease envelops a parabola with focus \(C\) and directrix \(AB\).
Prove Pascal's theorem that the meets of opposite sides of a hexagon inscribed in a conic are collinear. \(A, B, C, D\) are four points of a parabola. Lines are drawn through \(C, D\) parallel to the axis to meet \(DA, BC\) in \(H\) and \(K\) respectively. Show that \(HK\) is parallel to \(AB\).
Prove that the common chords of a conic and circle taken in pairs are equally inclined to the axes of the conic. The common chord of the ellipse \[ x^2/a^2 + y^2/b^2 = 1 \] and the circle of curvature at a point \(P\) passes through the point \((\xi, \eta)\). Show that there are four such points \(P\) and that they lie on the circle \[ 2(x^2+y^2) - (a^2-b^2)(\xi x/a^2 - \eta y/b^2) - a^2-b^2 = 0. \]
Prove that the extremities of parallel diameters of the circles of a coaxal system \(\Sigma\) lie on a rectangular hyperbola. Show further that the circles whose diameters are the chords of this hyperbola perpendicular to those of the first system form a second coaxal system orthogonal to \(\Sigma\).
A variable chord, \(PQ\), of a conic subtends a right angle at a fixed point \(O\). Show that the locus of the foot of the perpendicular from \(O\) on \(PQ\) is in general a circle. What is the condition that the locus should be a straight line?
A variable tangent to a conic \(S\) meets two fixed perpendicular tangents \(l,m\) at \(P, Q\) respectively, and the perpendiculars to \(l,m\) at \(P,Q\) meet in \(R\). Prove that, if \(S\) is a central conic, the locus of \(R\) is a rectangular hyperbola whose asymptotes are the tangents of \(S\) parallel to \(l,m\); and that this hyperbola passes through the points of contact of \(l\) and \(m\) with \(S\). What does the locus of \(R\) become if \(S\) is a parabola?
Prove that the locus of poles of a line \(l\) with respect to a system of confocal conics is a line \(m\) perpendicular to \(l\). If \(l\) meets a particular conic \(S\) of the system in \(P, Q\), prove that the normal to \(S\) at \(P\) meets \(m\) in the pole of \(l\) with respect to the other confocal through \(P\).
A straight line \(l\) meets the sides \(BC, CA, AB\) of a triangle in \(A_1, B_1, C_1\) respectively. \(O\) is a general point of the plane and \(A', B', C'\) are general points on the lines \(OA_1, OB_1, OC_1\) respectively. \(B'C'\) meets \(l\) in \(\alpha\), and \(\beta, \gamma\) are similarly defined. Prove that \(A\alpha, B\beta, C\gamma\) are concurrent.