In a triangle $ABC$,$CH$ and $CM$ are the lengths of the altitude and median to the base $AB$ respectively. If $a = 10, b = 26, c = 32$,then find the length of $HM$.

The common roots of the equations $2\sin^2 x + \sin^2 2x = 2$ and $\sin 2x + \cos 2x = \tan x$ are

The product of the arithmetic mean of the lengths of the sides of a triangle and the harmonic mean of the lengths of the altitudes of the triangle is equal to: [where $\Delta$ is the area of the triangle $ABC$]

Let a triangle $ABC$ be inscribed in a circle of radius $2$ units. If the $3$ bisectors of the angles $A, B$ and $C$ are extended to cut the circle at $A_1, B_1$ and $C_1$ respectively,then the value of $\left[\frac{AA_1 \cos \frac{A}{2} + BB_1 \cos \frac{B}{2} + CC_1 \cos \frac{C}{2}}{\sin A + \sin B + \sin C}\right]^2$ is:

If $A, B, C$ are the angles of a triangle,then $\frac{\sin A+\sin B+\sin C}{\sin ^2 \frac{A}{2}-\sin ^2 \frac{B}{2}+\sin ^2 \frac{C}{2}-1} =$

In any triangle $ABC$,the value of $a(b^2 + c^2)\cos A + b(c^2 + a^2)\cos B + c(a^2 + b^2)\cos C$ is