\(P\) is a point inside a quadrilateral \(ABCD\) such that the sum of the areas \(PAB, PCD\) is constant. Prove that the locus of \(P\) is a straight line.
Each edge of a tetrahedron \(OPQR\) is equal to the opposite edge, and \(A, B, C\) are inverse to \(P, Q, R\) with respect to \(O\). Prove that the foot of the perpendicular from \(O\) on the plane \(ABC\) is the incentre \(I\) of the triangle \(ABC\).
\(ABCDE\) is a structure consisting of 7 equal light rods lying in a plane and freely jointed at their ends. It is supported at \(A\) and \(D\) so that \(AD\) is horizontal and it supports a load \(W\) at \(E\). Draw a force diagram and determine the stresses in the various members. Find the changes in the stresses in the various members when extra rods \(AC, BD\) are inserted in the positions shown by the dotted lines in the diagram and are so adjusted that they each take a thrust equal to \(W\). % Diagram of a 7-rod structure is present in the original document. % It shows a symmetrical roof-truss like structure. % Vertices are A, D on the bottom, horizontal. % From A, rods go to B and E. From D, rods go to C and E. % Rods connect B to C, B to E, and C to E. % The shape consists of three triangles: ABE, BCE, CDE. % ABE and CDE are congruent isosceles triangles. BCE is isosceles. % The load W is at E. Dotted lines show extra rods AC and BD.
Give a short account, without proofs, of the principal properties of the three transformations: (1) conical projection, (2) inversion, (3) reciprocation. Discuss the chief characteristics in which these transformations are alike or different, e.g. that in (1), (2), but not in (3), a point corresponds to a point, and that in (1) and (3), but not generally in (2), a conic corresponds to a conic.
\(A\) and \(B\) are the centres of two circles which intersect in \(P\) and \(Q\); the angle \(APB\) is less than a right angle. Prove that one or other of the angles between tangents at \(P\) to the circles \(APQ, BPQ\) is twice the angle \(APB\).
A variable straight line through the centre \(O\) of a regular hexagon \(ABCDEF\) meets \(AC\) in \(G\) and \(AE\) in \(H\). Prove that \(BG, FH\) meet on the circle circumscribed to the hexagon.
A uniform rectangular door of depth \(a\) weighing \(W\) lbs. slides in vertical grooves and is supported by a vertical chain which is attached at a distance \(c\) from the centre line of the door. The distance apart of the grooves is slightly greater than the width of the door, the coefficient of friction between the door and the grooves is \(\mu\) and \(\mu\) is less than \(a/2c\). Show that the difference between the tensions of the chain when the door is being raised and lowered slowly is \[ \frac{4a\mu c W}{a^2 - 4\mu^2 c^2}. \]
Find how many conics (not necessarily real) can be drawn to pass through \(m\) given points and touch \(5-m\) given straight lines, in the six cases when \(m = 5, 4, 3, 2, 1, 0\). It is assumed that no three given points are collinear, no three given straight lines concurrent, and that no given point lies on any of the given straight lines. Give geometrical constructions for the points of contact in the two cases \(m=4, 3\).
Find the factors of \[ \begin{vmatrix} a^2 & a^3 & a & 1 \\ b^2 & b^3 & b & 1 \\ c^2 & c^3 & c & 1 \\ d^2 & d^3 & d & 1 \end{vmatrix}. \]
A normal to an ellipse, of eccentricity \(1/\sqrt{2}\), at a point whose eccentric angle is \(\theta\), meets the ellipse again at a point whose eccentric angle is \(\phi\). Prove that \[ \tan\left(\frac{1}{4}\pi - \frac{1}{2}\theta\right) \tan\left(\frac{1}{4}\pi - \frac{1}{2}\phi\right) = -1. \]