Prove that the inverse of a circle is either a circle or a straight line. Two fixed circles \(C\) and \(C'\), of radii \(r\) and \(r'\), touch at the point \(O\). Two variable circles \(\Gamma\) and \(\Gamma'\) each touch both \(C\) and \(C'\), and also touch each other at the point \(P\). Show that as \(\Gamma\) and \(\Gamma'\) vary, \(P\) describes a circle touching \(C\) and \(C'\) at \(O\). Find the radius of this circle in terms of \(r\) and \(r'\), distinguishing between the cases in which \(C\) and \(C'\) are on the same or on opposite sides of their common tangent.
Deduce from the triangle (or parallelogram) of forces (i) that a system of forces in a plane can be reduced either to a single force or to a couple, (ii) that the sum of the moments of the forces about any point in the plane is equal to the moment of this resultant force about the point or, if the system reduces to a couple, to the moment of this couple.
Two conics \(S\) and \(S'\) meet in the four points \(A, B, C, D\). Through \(A\) a variable line \(l\) is drawn meeting \(S\) in \(P\) and \(S'\) in \(X\). \(BP\) meets \(S'\) again in \(Y\), and \(BX\) meets \(S\) again in \(Q\). Show that if \(A, Y, Q\) are collinear for one position of \(l\), then this will also be the case for every position of \(l\), and as \(l\) varies, the lines \(PQ\) all pass through a fixed point \(U\), and the lines \(XY\) all pass through a fixed point \(V\). Show further that in this case the pole of \(AB\) with respect to \(S\) is the same as the pole of \(CD\) with respect to \(S'\), and the pole of \(CD\) with respect to \(S\) is the same as the pole of \(AB\) with respect to \(S'\).
Prove that, if \((1 + x)^n = c_0 + c_1x + \dots + c_nx^n\), then \[c_0 c_2 + c_1 c_3 + \dots + c_{n-2}c_n = \frac{(2n)!}{(n-2)!(n+2)!},\] and \[\frac{c_0}{2} + \frac{c_1}{3} + \dots + \frac{c_n}{n+2} = \frac{2^{n+1}n+1}{(n+1)(n+2)}.\]
Two pencils, with vertices \(A\) and \(B\), are homographically related in such a way that the ray \(AB\) of the first corresponds to the ray \(BA\) of the second. Prove that the locus of the points common to two corresponding rays consists of the line \(AB\) together with a second line \(l\). Two homographic ranges lie on the same line. Show that there are two self-corresponding points, which may be either real and distinct, or coincident, or imaginary, and one or both of which may be at infinity. If the self-corresponding points are real, show that the two ranges are in perspective (from different centres) with one and the same range of points on a second suitably chosen line \(l\).
Four equal uniform straight rods \(AB, BC, CD, DE\), each of length \(2a\) and weight \(W\), are smoothly jointed at \(B, C\) and \(D\). The ends \(A\) and \(E\) are freely hinged to fixed points \(2a \operatorname{cosec}(\pi/8)\) apart at the same level. Show that the rods can be made to hang in the form of the lower half of a regular octagon by applying two equal and opposite couples, either to \(AB\) and \(DE\) or to \(BC\) and \(CD\). Prove that the ratio of the couples needed in the two cases is \(-\cot(\pi/8)\).
Two coplanar triangles \(ABC\) and \(A'B'C'\) are in perspective from a point \(O\). Prove that, of the nine points where a side of \(ABC\) meets a side of \(A'B'C'\), three are collinear and the remaining six lie on a conic.
Prove that, if \[ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy\] is the product of two factors linear in \(x, y\) and \(z\), then \[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0. \] Prove that, if \(A, B\) and \(C\) are the angles of a triangle, \[ \begin{vmatrix} \sin 2A & \sin C & \sin B \\ \sin C & \sin 2B & \sin A \\ \sin B & \sin A & \sin 2C \end{vmatrix} = 0. \]
\(X\) and \(Y\) are any points of the line \(AB\), and \(X'\), \(Y'\) are their harmonic conjugates with respect to \(A, B\). Prove that \((ABXY)=(ABX'Y')\). \(P\) and \(Q\) are two points coplanar with a conic \(S\). The tangents from \(P\) to \(S\) have points of contact \(A\) and \(B\). The tangents from \(Q\) to \(S\) have points of contact \(C\) and \(D\). Show that the six points \(P, Q, A, B, C, D\) lie on a conic.