Properties of Triangle

Let $\left(5, \frac{a}{4}\right)$ be the circumcenter of a triangle with vertices $\mathrm{A}(a,-2), \mathrm{B}(a, 6)$ and $C\left(\frac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha+\beta+\gamma$ is

Two vertices of a triangle $\mathrm{ABC}$ are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$, and its orthocentre is $\mathrm{P}(1,1)$. If the coordinates of the point $\mathrm{C}$ are $(\alpha, \beta)$ and the centre of the of the circle circumscribing the triangle $\mathrm{PAB}$ is $(\mathrm{h}, \mathrm{k})$, then the value of $(\alpha+\beta)+2(\mathrm{~h}+\mathrm{k})$ equals

In a triangle with sides $a, b, c,$ ${r_1} > {r_2} > {r_3}$ (which are the ex-radii) then :

The sum of the radii of inscribed and circumscribed circles for an $n$ sided regular polygon of side $a,$ is :

If in a $\Delta ABC$ $a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2},$ then the sides $a, b$ and $c$ :

For a regular polygon, let $r$ and $R$ be the radii of the inscribed and the circumscribed circles. A $false$ statement among the following is :

In a triangle $ABC$, medians $AD$ and $BE$ are drawn. If $AD=4$, $\angle DAB = {\pi \over 6}$ and $\angle ABE = {\pi \over 3}$, then the area of the $\angle \Delta ABC$ is :

A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (–1, 1) and (2, 3). Then the centroid of this triangle is :

Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be $\left( {{{10} \over 3},{7 \over 3}} \right)$. If $\alpha$, $\beta$ are the roots of the equation $a{x^2} + bx + 1 = 0$, then the value of ${\alpha ^2} + {\beta ^2} - \alpha \beta$ is :

The angles A, B and C of a triangle ABC are in A.P. and a : b = 1 : $\sqrt 3$. If c = 4 cm, then the area (in sq. cm) of this triangle is :