Points \(X, Y, Z\) are taken in the sides \(BC, CA, AB\) of an equilateral triangle \(ABC\) and \(AX, BY, CZ\) form the triangle \(PQR\). Prove that, if the triangle \(PQR\) is equilateral, so also is the triangle \(XYZ\).
Prove that the common tangents to two circles whose centres are \(A\) and \(B\) cut the line \(AB\) in the points which divide \(AB\) internally and externally in the ratio of the radii of the circles. Prove also that the other points in which the common tangents intersect each other lie on the circle whose diameter is \(AB\).
The tangents to the circumcircle of a triangle \(ABC\) cut the opposite sides in \(X, Y, Z\). Prove the circles whose diameters are \(AX, BY, CZ\) have a common radical axis which passes through the orthocentre and the circumcentre of the triangle.
Prove that the curve of intersection of two spheres is a circle in a plane perpendicular to the line that passes through their centres. Prove that the mid-points of the edges of a cube that do not pass through either end of a given diagonal are the vertices of a regular hexagon which lies in the plane which bisects the given diagonal at right angles.
Prove that the circumcircle of the triangle formed by three tangents to a parabola passes through the focus. The tangents to a parabola at \(P\) and \(Q\) meet in \(T\), and the tangent that is parallel to \(PQ\) cuts \(TP, TQ\) in \(Y, Z\). Prove that, if the circle \(TYZ\) cuts the diameter through \(T\) in \(H\), then the line joining \(H\) to the focus is parallel to \(PQ\).
Prove that, if any chord \(PQ\) of a hyperbola cuts the asymptotes in \(M, N\), then \(MP = QN\). Having given three points \(A, B, C\) on a hyperbola and one asymptote, shew how to construct (1) the other asymptote, and (2) the tangent at \(A\).
Prove that \(a+b-c-d\) is a factor of \[ (a+b+c+d)^3 - 6(a+b+c+d)(a^2+b^2+c^2+d^2) + 8(a^3+b^3+c^3+d^3); \] and resolve this expression into its simple factors.
Prove that, if \(s_n = \alpha^n+\beta^n\), where \(\alpha, \beta\) are the roots of \(x^2-ax+b=0\), \[ s_n = a s_{n-1} - b s_{n-2}. \] Hence, or otherwise, prove that if \(u_n\) denotes the expression \[ a^n s_n + n b a^{n-1} s_{n-1} + \frac{n(n+1)}{1.2} b^2 a^{n-2} s_{n-2} + \dots + \frac{(2n-2)!}{(n-1)!(n-1)!} b^{n-1} a s_1, \] then \[ u_n = a^n u_{n-1} = a^{2n}. \]
Prove that, when \(b-a\) is small compared with \(a\) the expression \(\log_e(b/a)\) is approximately equivalent to \(2(b-a)/(b+a)\). Shew that, if \(b-a\) does not exceed \(a/10\), the error in using this approximation is less than \(.0001\).
A weight of 20 oz. is supported by two strings one of which is tied to a fixed point \(A\) while the other passes over a smooth peg at \(B\) in the same horizontal line as \(A\) and has its other end attached to a weight of 7 oz. If the length of the former string and the distance \(AB\) are each 10 inches, determine the depth of the 20 oz. weight below \(AB\) in the position of equilibrium.