In any $\triangle ABC$,the simplified form of $\frac{\cos 2A}{a^2} - \frac{\cos 2B}{b^2}$ is

In a triangle,the lengths of the sides are integers. Suppose that the length of one side is $1$,and the longest altitude is twice the shortest altitude. Let $R$ and $r$ be the circumradius and inradius of the triangle,respectively. If $R:r = m:n$,where $m$ and $n$ are coprime positive integers,then $m + n$ is

With usual notations,if in $\triangle ABC$,$s$ is the semi-perimeter and $(s-a)(s-b)=s(s-c)$,then $\triangle ABC$ is

In a $\triangle ABC$,the expression $\frac{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}{4b^2c^2}$ equals:

In a triangle $ABC$,if $\tan \left(\frac{A-B}{2}\right) = \frac{1}{3} \tan \left(\frac{A+B}{2}\right)$,then $a : b =$

If angles $A, B$ and $C$ are in $A$.$P$.,then $\frac{a+c}{b}$ is equal to