Upon a given line as base and upon the same side of it six triangles may be constructed equiangular to a given triangle. Prove that the six vertices of these triangles lie on a circle, and that all such circles belong to a coaxal system.
Four equal smooth cylinders of weight \(W\) are placed inside another cylinder as shewn in the diagram, all axes being horizontal. Assuming that the reaction at the point of contact \(M\) vanishes, shew that the reaction between the cylinders, centres \(A\) and \(B\), is \[ \frac{1}{\sqrt{2}}W (3 \cos \theta + \sin \theta). \] [Diagram showing a large circle containing four smaller circles of equal size. The two lower circles have centers A and B. A vertical line from the center of the large circle passes between A and B, labeled W with a downward arrow at the bottom. An angle \(\theta\) is indicated.]
Define the polar of a point with respect to a circle, and show that if \(P\) lies on the polar of \(Q\) then \(Q\) will lie on the polar of \(P\). Show also that the circle of which \(PQ\) is a diameter will intersect the former circle at right angles.
Write an account of the method of inversion, giving a general sketch of the method rather than rigorous proofs. Consider in particular the effect of inversion upon a system of coaxal circles and upon a pair of points mutually inverse with respect to a circle. Prove the two following theorems:
The 1000 yards range trajectory of a rifle bullet is given by the following heights (in feet) above the ground at intervals of 100 yards: 0, 7.3, 13.7, 18.9, 22.8, 25.0, 25.1, 23.1, 18.4, 10.9, 0. Deduce the figures which specify in the same way the trajectory for 500 yards range.
Two triangles \(\Delta\) and \(\Delta'\) are inscribed in the same circle, and in each a vertex and the side opposite to it are selected. From the selected vertex \(X\) of \(\Delta\) a chord \(XP\) of the circle is drawn parallel to the selected side of \(\Delta'\), and from the selected vertex \(X'\) of \(\Delta'\) a chord \(X'P'\) is drawn parallel to the selected side of \(\Delta\). Prove that the length of the chord \(PP'\) is the same whatever vertices are selected.
\(AB, CD\) are segments of two fixed coplanar lines. If \(AB\) be of fixed length and likewise \(CD\), shew that the resultant of forces represented by \(AC, DB\) has a fixed direction and magnitude.
Prove that the tangents drawn from a point to an ellipse subtend equal angles at a focus. Under what circumstances does the theorem need modification in the case of a hyperbola? Prove that if \(P\) and \(Q\) are two points of a conic whose foci are \(S, S'\), a circle can be drawn which touches \(SP, SQ, S'P, S'Q\).
Prove that the sine, cosine and tangent of any multiple of \(\theta\) are rational algebraic functions (i) of \(\tan\frac{1}{2}\theta\), (ii) of \(\cos\theta + i\sin\theta\). In what way are these facts applied to the solution of trigonometrical equations and to the integration of trigonometrical functions? Shew that the equation \[ \cos(2\theta - \alpha) = k \cos(\theta - \beta) \] can be satisfied by four values \(\theta_1, \theta_2, \theta_3, \theta_4\) of \(\theta\), of which no two differ by a multiple of \(2\pi\), and by no more. Shew also that \[ \theta_1 + \theta_2 + \theta_3 + \theta_4 - 2\alpha = 2n\pi, \] and that \[ \cos(\theta_2 + \theta_3 - \alpha) + \cos(\theta_3 + \theta_1 - \alpha) + \cos(\theta_1 + \theta_2 - \alpha) = 0. \]
A circular hill is very nearly of the form given by a regular truncated cone 3000 ft. in diameter at the base and 1000 ft. in diameter at the top, the height being 40 ft. A railway cutting is made through the hill, the bottom of the cutting being level with the base and 25 ft. wide, whilst the width on the flat top of the hill is 100 ft. Estimate the volume of earth to be excavated.