Relations Between Sides and Angles of a Triangle

Find the maximum perimeter of a triangle on a given base  $\text{'}a\text{'}$and having the given vertical angle $\alpha$.

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

If in a triangle $\mathrm{ABC},\mathrm{cosA}+\mathrm{cosB}=4{\sin }^2\frac{\mathrm{C}}{2}$, show that its sides are in A.P.

If in $∆\mathrm{ABC},\mathrm{cosA}+2\mathrm{cosB}+\mathrm{cosC}=2$ then show that the sides of the triangle are in A.P.

If in a triangle, the casines of two angles are proportional to the opposite sides then show that the triangle is an isosceles triangle.

If in $\Delta \mathrm{ABC},\frac{1}{a+c}+\frac{1}{b+c}=\frac{3}{a+b+c}$, then show that $\mathrm{C}=60°$.

In a triangle $\mathrm{ABC},a=2b$ and $\angle \mathrm{A}=3\angle \mathrm{B}$ show that the triangle is right-angled.

The sides of a triangle are $2x+3,x^2+3x+3,x^2+2x$, prove that the greatest angle is $\frac{2\pi }{3}$.

In a triangle $\mathrm{ABC}$, if $b^2=a(c+a),c^2=b(a+b)$ then prove that $\cos A\cos B\cos C=-\frac{1}{8}$

Show that, $a\cos B\cos C+b\cos C\cos A+c\cos A\cos B=\frac{abc}{4R^2}$

If in a triangle $\mathrm{ABC},a^4+b^4+c^4=2c^2\left(a^2+b^2\right)$ find $\mathrm{C}$.

If in a triangle $\mathrm{ABC},\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$, then prove that $\frac{\cos A}{7}=\frac{\cos B}{19}=\frac{\cos C}{25}$.

In a $∆\mathrm{ABC}$ prove that $\frac{\cos B\cos C}{bc}+\frac{\cos C\cos A}{ca}+\frac{\cos A\cos B}{ab}=\frac{{\sin }^{2}A}{a^2}$.

Prove that in any triangle $ABC,\frac{a+b}{c}=\frac{\cos \frac{1}{2}( \mathrm{A}-\mathrm{B})}{\sin \frac{1}{2}\mathrm{C}}$ and $\frac{a-b}{c}=\frac{\sin \frac{1}{2}( \mathrm{A}-\mathrm{B})}{\cos \frac{1}{2}\mathrm{C}}$.

If in a triangle $\mathrm{ABC}$, the angles are in A.P., show that $2\cos \frac{\mathrm{A}-\mathrm{C}}{2}=\frac{a+c}{\sqrt{a^2+c^2-ac}}$

In a triangle $\mathrm{ABC}$, prove that if $a^2,b^2,c^2$ are in A.P., then $\mathrm{cotA},\mathrm{cotB}$ and $\mathrm{cotC}$ are in A.P.

Three sides of a triangle are $4,5$ and $6 \mathrm{cm}$ respcctively. Show that the least angle is half the greatest.

In a triangle $\mathrm{ABC},\mathrm{A}=60°$ and $b:c=(\sqrt{3}+1):2$. Find the angles $\mathrm{B}$ and $\mathrm{C}$.

[xii] If $\mathrm{C}=60°$ then $2a-b=2c\mathrm{cosB}$