Area of Triangle

Consider points inside a $∆$ from which the sum of the squares of distance to the three sides is minimum. If the minimum value of the sum of square of distances is $M$, then find$\frac{a^2+b^2+c^2}{(s-a)(s-b)(s-c)s}\times M$ .

If $A$ is the area and $2s$ the sum of three sides of a triangle, then

In a triangle $ABC$, the length of the bisector of angle $A$ is

The sides of a triangle are given by $\sqrt{b^2+c^2}, \sqrt{c^2+a^2}, \sqrt{a^2+b^2}$. Where $a, b, c>0,$ then the area of the triangle equals

A given chord $AB$ of a given circle subtends an angle $\text{θ}$ at a point $C$ on the circumference, triangle $ABC$ has maximum area when:

The sides of a $\Delta$ are in $\text{A.P.}$ and its area is $\frac{3}{5}\times$ (area of an equilateral triangle of the same perimeter). Find the ratio of its sides.

In a $\text{Δ}ABC, B=90°, AC=h$ and the length of perpendicular from $B$ to $AC$ is $p$ such that $h=4 p.$ If $AB<BC$, then measure $\angle C$.

Find the value of $\tan A,$ if area of $\text{Δ}ABC$ is $a^2-{\left(b-c\right)}^2$.

In $\text{Δ}ABC, A=\frac{2\text{π}}{3}, b-c=3\sqrt{3} \mathrm{cm}$ and $\mathrm{area}\left(\text{Δ}ABC\right)=\frac{9\sqrt{3}}{2}{\mathrm{cm}}^2.$ Solve for side $a$.

In a $\text{Δ}ABC$, $a, b \& A$ are given and $c_1 \& c_2$ are two values of the third side $c$. Find the sum of the areas of the two triangles with side lengths $a,b \& c_1$ and $a,b \& c_2$.

Consider a $∆ABC$. A directly similar $∆A_1{B}_1{C}_1$ is inscribed in the $∆ABC$ such that $A_1, B_1$ and $C_1$ are the interior points of the sides $AC, AB$ and $BC$ respectively. Prove that:
$\frac{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\left(\Delta A_1{B}_1{C}_1\right)}{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\left(\Delta ABC\right)}\ge \frac{1}{{\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}}^{2}A+{\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}}^{2}B+{\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}}^{2}C}$

In $∆ABC$, ‘ $h$’ is the length of altitude drawn from vertex $A$ on the side $BC$.
Prove that:
$2\left(b^2+c^2\right)\ge 4h^2+a^2$. Also discuss the case when equality holds true.