If the incentre of the triangle formed by the lines $x-2=0$,$x+y-1=0$,and $x-y+3=0$ is $(\alpha, \beta)$,then $\beta=$

Suppose $ABOC$ is a rhombus in the first quadrant with $O$ being the origin. If the vertices $B$ and $C$ of $\triangle ABC$ lie respectively on $y=\frac{4}{3}x$ and $y=0$,and the side $BC$ passes through $\left(\frac{2}{3}, \frac{2}{3}\right)$,then the mid-point of $BC$ is

In a $\triangle ABC$,points $X$ and $Y$ are on $AB$ and $AC$,respectively,such that $XY$ is parallel to $BC$. Which of the two following equalities always hold? (Here $[PQR]$ denotes the area of $\triangle PQR$).
$I$. $[BCX] = [BCY]$
$II$. $[ACX] \cdot [ABY] = [AXY] \cdot [ABC]$

Suppose $\triangle ABC$ is an isosceles triangle with $\angle C=90^{\circ}$,$A=(2,3)$ and $B=(4,5)$. Then the centroid of the triangle is

The points $(0, 0)$,$(a, 0)$,and $\left( \frac{a}{2}, \frac{a\sqrt{3}}{2} \right)$ are vertices of:

If a straight line $L$ is perpendicular to the line $4x - 2y = 1$ and forms a triangle of area $4 \text{ sq unit}$ with the coordinate axes,then the equation of the line $L$ is