The equations of sides $AB$,$BC$,and $CA$ of a $\triangle ABC$ are $2x+y=0$,$x+py=q$,and $x-y=3$ respectively. If $P(2,3)$ is its orthocenter,then the value of $p+q$ equals

Let $B$ and $C$ be two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on the line $y-2x=2$ such that $\triangle ABC$ is an equilateral triangle. Then,the area of the $\triangle ABC$ is

Two equal sides of an isosceles triangle are given by $7x-y+3=0$ and $x+y-3=0$. If the slope $m$ of the third side is an integer,then $m=$

The coordinates of the vertices of a quadrilateral are $(2, -1)$,$(0, 2)$,$(2, 3)$,and $(4, 0)$. The angle between its diagonals is:

$A$ line $4x+y=1$ passes through the point $A(2,-7)$ and meets the line $BC$,whose equation is $3x-4y+1=0$,at the point $B$. The equation of the line $AC$ such that $AB=AC$ is

Let $m_{1}, m_{2}$ be the slopes of two adjacent sides of a square of side $a$ such that $a^{2}+11 a+3(m_{1}^{2}+m_{2}^{2})=220$. If one vertex of the square is $(10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))$,where $\alpha \in(0, \frac{\pi}{2})$ and the equation of one diagonal is $(\cos \alpha-\sin \alpha) x +(\sin \alpha+\cos \alpha) y =10$,then $72(\sin ^{4} \alpha+\cos ^{4} \alpha)+a^{2}-3 a+13$ is equal to.