If the straight lines $x + 3y = 4$,$3x + y = 4$,and $x + y = 0$ form a triangle,then the triangle is

In an isosceles triangle $ABC$,the vertex $A$ is $(6,1)$ and the equation of the base $BC$ is $2x + y = 4$. Let the point $B$ lie on the line $x + 3y = 7$. If $(\alpha, \beta)$ is the centroid of $\triangle ABC$,then $15(\alpha + \beta)$ is equal to

The angular points of a triangle are $A(-1, -7)$, $B(5, 1)$, and $C(1, 4)$. The equation of the bisector of the angle $\angle ABC$ is

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 $L_1$ be a line passing through $(2,1)$ and $(3, \frac{5}{2})$. $L_2$ is a line perpendicular to $L_1$ and passing through $(4,-1)$. The area of the triangle formed by $L_1$,$L_2$ and the $y$-axis is

Points inside the triangle with vertices at $(1,3), (5,0)$ and $(-1,2)$ must necessarily satisfy which of the following conditions?