The internal bisectors of the angles \(A, B, C\) of a triangle meet the circumcircle in \(A', B', C'\). Prove that, if \(\Delta, \Delta'\) are the areas of the triangles \(ABC, A'B'C'\), \[ \frac{\Delta'}{\Delta} = \frac{R}{2r}. \] Prove also that, if \(B'C'\) meets \(AB, AC\) in \(Z\) and \(Y\), \[ AY = AZ = 2Rr/a, \] and from this and similar results shew that the diagonals of the convex hexagon formed by the sides of the triangles \(ABC, A'B'C'\) pass through the incentre of the triangle \(ABC\).
A vertical flagstaff \(AB\) is observed to subtend the same angle at two points \(P, Q\) at the same level in a plane through \(AB\). The elevations of the base of \(AB\) as seen from \(P, Q\) are \(\alpha, \beta\) (\(\alpha > \beta\)) and the distance between \(P\) and \(Q\) is \(a\). Prove that \[ AB = a \cos(\alpha + \beta)/\sin(\alpha - \beta). \]
Find all the values of \(\theta\) that satisfy the equation \[ \tan\theta \cot(\theta+\alpha) = \tan\beta \cot(\beta+\alpha). \] Eliminate \(\theta\) from \[ a \cos(\alpha - 3\theta) = 2b \cos^3\theta \quad \text{and} \quad a \sin(\alpha - 3\theta) = 2b \sin^3\theta. \]
Prove that \(\cos n\theta\) where \(n\) is an integer can be expressed as a rational function of \(\cos\theta\) of degree \(n\). Prove that, if \(2 \cos\theta = x\), \[ \frac{1+\cos 9\theta}{1+\cos\theta} = (x^4 - x^3 - 3x^2 + 2x + 1)^2. \]
In a triangle prove that
Expand \(\log(1+2h\cos\theta+h^2)\) in the form \(\sum A_n h^n \cos n\theta\) and find \(A_n\). If \[ y = \log\left\{\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\right\} = a_1 x + a_3 x^3 + a_5 x^5 + \dots, \] prove that \[ x = a_1 y - a_3 y^3 + a_5 y^5 - \dots . \]
Prove that in general any system of coplanar forces can be reduced to a single force acting through a given point together with a couple. A triangle \(ABC\) is formed of three light rods freely jointed to each other at their ends. The rods are acted on by coplanar forces at their middle points perpendicularly to the rods and respectively proportional to the rods. Prove that if the forces all act outwards the system is in equilibrium and that the stresses at the joints are all equal and act along the tangents at the angular points to the circumcircle of the triangle \(ABC\).
State the laws of statical friction and find the least force that will support a heavy particle in equilibrium when placed on an inclined plane whose inclination to the horizontal is greater than the angle of friction. A rectangular block on a square base of side \(a\) is of height \(b\) and rests on a rough inclined plane of inclination \(\alpha\) so that two faces are vertical. A string is attached to the middle point of the highest edge and is pulled up the plane in a direction parallel to a line of greatest slope. Prove that, as the tension of the string is gradually increased, the equilibrium of the block will be broken by sliding if \(\mu < (a-b\tan\alpha)/2b\), where \(\mu\) is the coefficient of friction between the block and the face of the plane.
Prove that, (i) the centre of inertia of a uniform triangular lamina is the same as that of three equal particles placed at the angular points, (ii) the centre of inertia of a uniform quadrilateral lamina is the same as that of four equal particles placed at the angular points and of an equal negative particle placed at the intersection of the diagonals. Find the ratio of the masses of three particles placed at the angular points of a triangle such that their centre of inertia is at the orthocentre of the triangle.
State the principle of conservation of linear momentum. A wedge of mass \(M\) whose faces are each inclined at an angle of 45° to the horizontal rests with its base on a smooth horizontal plane and is free to move in a direction perpendicular to its edge. Particles of masses \(m, m'\) connected by a light string passing over the edge are placed one on each face of the edge with the string taut. Prove that when the system is released from rest the acceleration of the wedge is \[ (m \sim m')g/(2M+m+m'). \]