Find the trilinear equation of the circle which circumscribes the fundamental triangle \(ABC\). Prove that the equation of the circle with centre \(A\) which touches \(BC\) is \[ \alpha\sin A (\alpha\sin A+2\beta\sin B+2\gamma\sin C) = (\beta\cos B-\gamma\cos C)^2. \]
Find \(\sin 18^\circ\), and prove that \(\sin 54^\circ - \sin 18^\circ = \frac{1}{2}\). Eliminate \(\theta, \phi\) from \[ \sin\theta+\sin\phi=a, \quad \cos\theta+\cos\phi=b, \quad \cot\theta+\cot\phi=c. \]
Express \(\tan n\theta\) in terms of \(\tan\theta\) when \(n\) is a positive integer. Prove that \[ \sum_{r=1}^{r=10} \operatorname{cosec}^2 \frac{r\pi}{11} = 40. \]
If \(r, R\) are the radii of the inscribed and circumscribed circles of the triangle \(ABC\) and \(s\) the semi-perimeter, prove that \[ r = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}, \quad s=4R\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}. \] Lines drawn parallel to the sides of the triangle \(ABC\) to touch the inscribed circle form with the sides of \(ABC\) three smaller triangles. Prove that the sum of the radii of the circles inscribed in these three triangles is equal to the radius of the circle inscribed in \(ABC\).
Sum the series:
Shew that forces represented in all respects by the lines joining any point to the angular points of a triangle have a resultant represented by three times the line joining the point to the centroid of the triangle. A light inextensible string is divided into six equal parts. The two ends are attached to the extremities of the horizontal diameter of a circle whose plane is vertical, and five particles are attached to the five points of division of the string. The length of the string and the weights of the particles are such that in equilibrium they lie on the circumference of the circle. Prove that their weights beginning at one end are in the ratios \[ \sqrt{3}+2:1:\sqrt{3}-1:1:\sqrt{3}+2. \]
Investigate the conditions of equilibrium of a rigid body acted on by any system of forces in a plane. Four uniform rods whose weight per unit length is \(w\) form a parallelogram \(ABCD\), freely jointed at its angles. It is suspended from the point \(A\), and \(A, C\) are joined by a string of such length that the figure is a rectangle. Find the tension of the string, and prove that the reactions at \(B\) and \(D\) are each equal to \[ w AB.BC/\sqrt{2AC}. \]
State the laws of Statical Friction. Find the least force that will just keep a heavy particle in equilibrium on a rough plane inclined to the horizontal at an angle greater than the angle of friction. A uniform heavy rod rests in limiting equilibrium with its two ends on the inside of a rough circular hoop placed in a vertical plane. The length of the rod is equal to the radius of the hoop and the highest end of the rod is at an extremity of the horizontal diameter of the hoop. Prove that the coefficient of friction between each end of the rod and the hoop is \((\sqrt{13}-2)/\sqrt{3}\).
State Newton's Laws of Motion. A smooth wedge of mass \(M\) and angle \(\alpha\) is free to move on a smooth horizontal plane in a direction perpendicular to its edge. A particle of mass \(m\) is projected directly up the face of the wedge with velocity \(V\). Prove that it returns to the point on the wedge from which it was projected after a time \[ 2V(M+m\sin^2\alpha)/\{(m+M)g\sin\alpha\}. \] Also find the pressure between the particle and the wedge at any time.
Prove that if the sum of the resolutes in a given direction of the external forces on any number of particles be zero, the sum of the momenta of the particles in that direction is constant. Two equal particles \(A\) and \(B\) are connected by a light inextensible string of length \(a\) which is stretched at full length perpendicular to the edge of the table. The particle \(A\) is drawn just over the edge of the table and is then released from rest in this position. Describe the nature of the subsequent motion and shew that after \(B\) leaves the table the centre of inertia of the two particles describes a parabola of latus rectum \(a/2\).