If the points $(1, 1)$,$(-1, -1)$ and $(-\sqrt{3}, k)$ are vertices of an equilateral triangle,then the value of $k$ will be:

The coordinates of the vertices $P, Q, R$ and $S$ of a square $PQRS$ inscribed in the triangle $ABC$ with vertices $A \equiv (0, 0)$,$B \equiv (3, 0)$,and $C \equiv (2, 1)$,given that two of its vertices $P$ and $Q$ lie on the side $AB$,are respectively:

Let the area of the triangle with vertices $A(1, \alpha)$,$B(\alpha, 0)$,and $C(0, \alpha)$ be $4 \text{ sq. units}$. If the points $(\alpha, -\alpha)$,$(-\alpha, \alpha)$,and $(\alpha^2, \beta)$ are collinear,then $\beta$ is equal to:

The points $(-a,-b), (a, b), (0,0)$ and $(a^{2}, ab)$ where $a \neq 0, b \neq 0$ are always

In an isosceles $\triangle ABC$,the coordinates of vertices $B$ and $C$ of the base $BC$ are $(3, 2)$ and $(2, 3)$ respectively. If the equation of the line $AB$ is $3y = 2x$,then the equation of the line $AC$ is

In a $\triangle ABC$ with $\angle A < \angle B < \angle C$,points $D, E, F$ are on the interior of segments $BC, CA, AB$ respectively. Which of the following triangles cannot be similar to $\triangle ABC$?