Two circles meet in the points \(A\) and \(B\) and tangents are drawn to them from a point \(P\) in their plane. Shew that if the four points of contact of the tangents are concyclic the point \(P\) must lie on the line \(AB\). Invert this theorem with respect to a circle whose centre is \(A\).
Shew that the two tangents to a conic which pass through a point are equally inclined to the lines which join the point to the foci of the conic. The points \(B\) and \(D\) are symmetrically placed with respect to \(AC\). Shew that the locus of the foci of conics which touch \(AB, BC, CD\) and \(DA\) is a circle and the line \(AC\).
Given three points \(A, B\) and \(C\) and two lines \(\alpha\) and \(\beta\) shew, by reciprocation and projection or otherwise, that there are in general four conics which pass through \(A, B\) and \(C\) and touch \(\alpha\) and \(\beta\). Shew further that the four chords of contact of these conics with \(\alpha\) and \(\beta\) form a quadrilateral whose diagonal triangle is \(ABC\).
Shew that if two coplanar triangles are in perspective from a point, called the centre of perspective, then the points of intersection of corresponding sides lie on a line, the axis of perspective. Three coplanar triangles are two by two in perspective. Shew that if they have the same centre of perspective the three axes of perspective are concurrent, and that if they have the same axis of perspective the three centres of perspective are collinear.
Consider three skew lines, \(a, b\) and \(c\), in space. \(A_1, A_2, A_3\) and \(A_4\) are four points on the line \(a\), and \(l_1, l_2, l_3\) and \(l_4\) are the lines through these points which meet \(b\) and \(c\). Shew that the cross ratio \((A_1 A_2 A_3 A_4)\) is equal to that of the four planes through \(a\) which contain the lines \(l_1, l_2, l_3\) and \(l_4\). (The cross ratio of four planes through a line \(\lambda\) is the cross ratio of the four points in which the planes meet any line which does not meet \(\lambda\).) By considering homographic ranges on \(a\), or otherwise, shew that, if \(d\) is another line, there are two lines which meet \(a, b, c\) and \(d\).
Obtain the equations of the two parabolas which pass through the points \((0,0), (7,0), (0,5),\) and \((3,-1)\), and the equations of their axes.
Shew that the orthocentre of the triangle formed by tangents to the parabola \(y^2 = 4ax\) at the points \((at_1^2, 2at_1), (at_2^2, 2at_2)\) and \((at_3^2, 2at_3)\) is the point \(\{-a, a(t_1+t_2+t_3) + at_1t_2t_3\}\). Normals are drawn to the parabola \(y^2=4ax\cos\alpha\) from any point on the line \(y=b\sin\alpha\). Shew that the orthocentre of the triangle formed by the tangents at the feet of the normals lies on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Four equal uniform freely jointed rods, forming a rhombus, rest in equilibrium with one diagonal vertical and the two lower rods supported on two smooth pegs at the same horizontal level. If \(a\) is the length of each rod, \(2c\) the distance between the pegs, and \(2b\) the horizontal diagonal of the rhombus, prove that \(b^3=a^2c\). If there is no reaction at the lowest joint, prove that \(8a=5\sqrt{5}c\).
Two rough uniform cylinders of equal radius rest in contact, with their axes horizontal, on a plane inclined at an angle \(\alpha\) to the horizon. If \(W_1\) is the weight of the upper cylinder and \(W_2\) of the lower, prove that \(W_1>W_2\), and that the coefficient of friction between the cylinders exceeds \((W_1+W_2)/(W_1-W_2)\). If \(\mu\) is the coefficient of friction between the plane and the upper cylinder, prove that \[ \tan\alpha < \frac{2\mu}{\mu+1} \cdot \frac{W_1}{W_1+W_2}. \]
A uniform thin hollow hemispherical bowl is in equilibrium on a horizontal plane with a smooth uniform straight rod resting partly within and partly without it. If the weight of the rod is half that of the bowl and its length is equal to the diameter of the spherical surface, prove that the inclination of the rod to the horizon is \[ \tan^{-1}\frac{1}{2}(2-\sqrt{2}). \]