Let $ABC$ be an equilateral triangle,and let $KLMN$ be a rectangle with $K, L$ on $BC$,$M$ on $AC$,and $N$ on $AB$. Suppose $AN / NB = 2$ and the area of $\triangle BKN$ is $6$. The area of the $\triangle ABC$ is:

If in a parallelogram $ABDC$,the coordinates of $A, B$ and $C$ are respectively $(1, 2), (3, 4)$ and $(2, 5)$,then the equation of the diagonal $AD$ is

The pair of lines $x^2 - 8x + 12 = 0$ and $y^2 - 14y + 45 = 0$ form a square. What is the center of the circle inscribed in the square?

$x+8y-22=0$, $5x+2y-34=0$, and $2x-3y+13=0$ are the three sides of a triangle. The area of the triangle is

Let $A, B$ be points on the two half-lines $x - \sqrt{3}|y| = \alpha, \alpha > 0$ at a distance of $\alpha$ from their point of intersection $P$. The line segment $AB$ meets the angle bisector of the given half-lines at the point $Q$. If $PQ = \frac{9}{2}$ and $R$ is the radius of the circumcircle of $\triangle PAB$, then $\frac{\alpha^2}{R}$ is equal to . . . . . .

$\Delta ABC$ has integral side lengths. The internal angle bisector of angle $B$ meets side $AC$ at $D$. If $AD = 3$ and $DC = 8$,then the perimeter of $\Delta ABC$ is: