Problems

\(BC\) is the hypotenuse of a right-angled triangle \(ABC\). Points \(D\) and \(E\) are taken in \(BC\) so that \(BD\), \(CE\) are equal to \(BA\), \(CA\), respectively, and \(F\) is the foot of the perpendicular from \(A\) to \(BC\). Show that \(AE\) bisects the angle \(BAF\) and that the angle \(DAE\) is half a right angle.

The reflexions of the vertices \(A, B, C\) of a triangle in the opposite sides are \(A'\), \(B'\), \(C'\). A triangle \(DEF\) is formed by drawing lines through \(A'\), \(B'\), \(C'\) parallel to \(BC\), \(CA\), \(AB\). Prove that the triangles \(DEF\), \(ABC\) have a common centroid (centre of gravity) and that their areas are in the ratio of 16 to 1.

Four light equal rods freely-jointed, are hung from fixed points \(A\) and \(B\) so that their vertices lie on the arc of a vertical circle centre \(O\).

The 2 lb. load and \(OC\) are vertical. Determine by a graphical construction the magnitude of \(F\) and \(G\), and the reactions at \(A\) and \(B\).

Give an account of various methods of finding the sum of \(n\) terms of series of the form \(\sum a_n, \sum a_n x^n\), (1) when \(a_n\) is a polynomial in \(n\), (2) when the coefficients are connected by a relation of the form \[ a_{n+k} + p_1 a_{n+k-1} \dots + p_k a_n = 0, \] \(p_1, p_2 \dots\) being constants, i.e. independent of \(n\). To what extent do the two classes of series (1) and (2) overlap? Find \(1^4+2^4 \dots + n^4\); and shew without carrying out the work in detail how the sum can be found by each of your other methods as far as applicable.

A chord of a circle cuts two fixed parallel chords so that the rectangles contained by the segments of the latter chords are equal. Show that its middle point lies on a straight line.

Two circles lie in different planes which meet in a straight line \(L\). Tangents \(PT\), \(PT'\) from a point \(P\) on \(L\) to the two circles are equal. Prove that, if the same property holds for a second point \(Q\) on \(L\), it holds for every point on \(L\).

Forces are represented by the sides of a plane polygon taken in order; show that they are equivalent to a couple the moment of which is represented by twice the area of the polygon. Show also that, if the polygon is not plane, forces represented by the sides taken in order are still equivalent to a couple. \(OA, OB, OC\) are lines mutually at right angles to each other. Show that forces represented by \(OA, AB, BC, CO\) are equivalent to a couple of moment \(OB.CA\).

Pairs of points \((P,P'), (Q,Q'), \dots\) on a straight line are in involution:

1. [(1)] if the cross-ratio of every set of four points is equal to the cross-ratio of the corresponding four points;
2. [(or 2)] if there is a fixed point \(O\) on the line such that \(OP.OP' = OQ.OQ' = \dots\);
3. [(or 3)] if there are two fixed points \(F_1, F_2\) on the line such that \(F_1, F_2\) is divided harmonically by each pair of points \((P,P'), \dots\);
4. [(or 4)] if the abscissae \(x, x'\) of each pair of points are connected by an equation \(axx'+h(x+x')+b=0\), \(a, h, b\) being constants.

Starting with any of these four properties (as you please), shew that the other three can in general be deduced, pointing out exceptional cases if any. Consider in particular the involutions defined by \[ xx'+1=0, \quad x+x'=0, \quad xx' - x - x' + 1 = 0. \] Give a geometrical construction, when possible, for the points \(O, F_1, F_2\) in the case of an involution determined by the intersections of a given straight line with a family of coaxal circles of either type.

Show that the coefficient of \(x^{3n+1}\) in the expansion of \(\displaystyle\frac{8-2x}{(x+2)(x^2+8)}\) in a series of ascending powers of \(x\) is \[ (-1)^{n+1} \frac{3n+3}{2^{3n+3}}. \]

Prove that, if \(A, B\) are ends of the axes of an ellipse, the circle on \(AB\) as diameter touches the ellipse, provided that the eccentricity is given by \[ e^2 = 2(\sqrt{2}-1). \]