\(S\) is the inscribed circle of a triangle \(A_1 A_2 A_3\), and \(S_1\), \(S_2\), \(S_3\) are the three escribed circles. Prove that the fourth common tangents of \(S\), \(S_1\) and of \(S_2\), \(S_3\) are parallel. Prove also that the triangle formed by the fourth common tangents of \(S\), \(S_1\), of \(S\), \(S_2\) and of \(S\), \(S_3\) is similar to the triangle whose vertices are the feet of the altitudes of the triangle \(A_1 A_2 A_3\).
A line segment \(AB\) of constant length \(b\) is such that \(A\) lies on the line \(y = 0\) while \(AB\) (produced if necessary in either direction) touches the circle \(x^2 + y^2 = a^2\). Find the equation of the locus of \(B\).
\(A\), \(B\), \(C\), \(D\) are four points on a parabola. The diameter through \(B\) meets \(CD\) in \(E\), and the diameter through \(D\) meets \(AB\) in \(F\). Prove that \(AC\) is parallel to \(EF\).
Show that the centre of curvature of the parabola \(y^2 = 4ax\) at the point \((at^2, 2at)\) is \([a(t^2 + 3t^2), -2at^3]\). The normal to a parabola at a point \(P\) meets the curve again in \(Q\). The tangents at \(P\) and \(Q\) meet in \(T\), and \(C\) is the centre of curvature at \(P\). Show that the mid-point of \(CT\) lies on the line through \(P\) perpendicular to the axis of the parabola.
\(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\) are eight points on an ellipse, such that the quadrangles \(ABCD\), \(ABEF\), \(CDGH\) are concyclic. Prove that \(EFGH\) are concyclic.
If \(A_1\), \(A_2\), \(A_3\), \(A_4\) are four points of a rectangular hyperbola, the normals at which are concurrent, show that each of these points is the orthocentre of the triangle formed by the other three, and that the conic through \(A_1\), \(A_2\), \(A_3\), \(A_4\) and the centre of the hyperbola is a second rectangular hyperbola whose axes are parallel to the asymptotes of the first.
Find the asymptotes of the plane cubic curve $$(a_1 x + b_1 y + c_1)(a_2 x + b_2 y + c_2)(a_3 x + b_3 y + c_3) + ax + by + c = 0,$$ assuming that no two of the lines \(a_i x + b_i y + c_i = 0\) are parallel. Show that, if the three finite points of intersection of the asymptotes with the curve coincide in \(O\), the curve has an inflexion at \(O\), and that any chord of the curve through \(O\) is bisected at \(O\). Sketch the curve \(y(x^2 - y^2) - x = 0\).
A point \(O\) is an origin of position vectors and \(P\), \(Q\), \(R\) are three distinct collinear points. Prove that the position vector \(\mathbf{OR}\) of the point \(R\) is of the form $$\frac{\mathbf{OP} + \lambda \mathbf{OQ}}{1 + \lambda},$$ with suitable choice of \(\lambda\). Four spheres \(S\), \(S_1\), \(S_2\), \(S_3\) touch each other externally in pairs. \(S\), \(S_1\) touch at \(P_{11}\); \(S_2\), \(S_3\) at \(P_{23}\); \(S_3\), \(S_1\) at \(P_{31}\) and \(S_1\), \(S_2\) at \(P_{12}\). Prove that the three lines \(P_1 P_{23}\), \(P_2 P_{31}\), \(P_3 P_{12}\) are concurrent. [Hint. Take the centre \(O\) of \(S\) as origin and express the position vectors of a general point of the line \(P_1 P_{23}\) in terms of \(\mathbf{OP_1}\), \(\mathbf{OP_2}\), \(\mathbf{OP_3}\).]
The diagonals \(A_1 A_3\), \(A_2 A_4\) of a quadrangle \(A_1 A_2 A_3 A_4\) intersect at right angles at \(O\). The line through \(O\) perpendicular to \(A_1 A_2\) meets \(A_1 A_2\) at \(P_{12}\) and \(A_3 A_4\) at \(Q_{12}\). The points \(P_{23}\), \(Q_{23}\), \(P_{34}\), \(Q_{34}\), \(P_{41}\), \(Q_{41}\) are similarly defined. Prove that the eight points \(P_{12}\), \(P_{23}\), \(P_{34}\), \(P_{41}\), \(Q_{12}\), \(Q_{23}\), \(Q_{34}\), \(Q_{41}\) are concyclic, and that \(Q_{12} Q_{23} Q_{34} Q_{41}\) is a rectangle.
A circle \(S\) rolls once round an equal circle \(S'\). Determine the area contained within the closed curve traced out by the point \(P\) of \(S\) which was first in contact with \(S'\).