Through the intersection of the diagonals of a quadrilateral lines are drawn parallel to the four sides. Shew that the four points in which they meet the sides, opposite to those to which they are parallel, lie on a straight line.
A uniform plank, 4 feet long, rests on a table with 9 inches projecting over the edge. An equal plank is placed on the first and projects 6 inches further. If a gradually increasing vertical force is applied downwards at the projecting end of the second plank, determine how equilibrium will be broken.
\(ABC\) is a triangle inscribed in a circle, the tangents at \(B\) and \(C\) meet at \(T\). Shew that, if a line through \(T\) parallel to the tangent at \(A\) meets \(AB, AC\) at \(D\) and \(E\) respectively, \(DE\) is bisected at \(T\).
Obtain projective generalisations of the following ideas: middle point of a line, bisector of an angle, right angle, angle of constant magnitude, circle, centre of a circle, concentric circles. Generalise by projection a few of the standard propositions involving these ideas and proved in the Third Book of Euclid's Elements.
A curve is drawn on a cone of vertical angle \(10^\circ\), such that any short portion of it may be made a straight line by developing the cone into a plane; shew that the curve intersects itself in two and only two points, and find the ratio of the distances of these points from the vertex of the cone. Also give the angles of intersection.
The pairs of points \((R, P'; P, P'; \dots)\) and the pairs of points \((P', P''; P_1, P_1''; \dots)\) form two involutions on the same straight line. Shew that the necessary and sufficient condition that the pairs of points \((P_1, P_1''; P_2, P_2''; \dots)\) should also form an involution is that the double points of the two first involutions should form a harmonic range.
Three coplanar forces are completely represented in magnitude and position by lines \(AA'\), \(BB'\), \(CC'\). Shew that their resultant is represented in magnitude and direction by \(3.GG'\), where \(G\) and \(G'\) are the centroids of the triangles \(ABC\) and \(A'B'C'\).
A point \(A\) moves along a straight line \(a\) and is joined to two fixed points \(B\) and \(C\) such that \(BC\) is parallel to \(a\). \(CA, BA\) meet a given straight line parallel to \(a\) in points \(E, F\), shew that the intersection of \(BE, CF\) describes a straight line.
Discuss the solution of the equations \[ ax+by+cz=d, \quad a'x+b'y+c'z=d', \quad a''x+b''y+c''z=d''. \] Distinguish carefully with numerical examples the cases in which there are an infinity of solutions, one, or none; and obtain conditions, in terms of the coefficients, sufficient to discriminate between the different cases. Interpret the results geometrically in terms of planes in space.
An aeroplane is flying at a uniform height at 100 ft. per sec. At a given instant an anti-aircraft gun range-finder registers the range as 9000 feet and the elevation as \(30^\circ\), whilst at that instant the direction of flight makes an angle of \(45^\circ\) with the vertical plane containing the gun and the aeroplane (the latter is approaching the gun). Neglecting the curvature of the trajectory and assuming that the average velocity of the projectile is 2000 ft. per sec., and that an interval of 10 seconds elapses between the registering of the range and the firing of the shot, at which instant the gun is sighted on the aeroplane, calculate approximately the correction which should be applied to the sight (1) in range, (2) in vertical elevation angle, (3) in horizontal training angle.