If $P$ is a point equidistant from all the vertices $A(-1, 3)$,$B(3, 5)$,and $C(5, 7)$ of a triangle $ABC$,then $PA=$

If the equations of the perpendicular bisectors of the sides $AB$ and $AC$ of a $\triangle ABC$ are $x-y+5=0$ and $x+2y=0$ respectively,and if $A$ is $(1,-2)$,then the equation of the perpendicular bisector of the side $BC$ is

In $\triangle ABC$,if $A=(1,2)$ and the equations of the medians through $B$ and $C$ are $x+y=5$ and $x=4$ respectively,then the area of $\triangle ABC$ is

If $(a, b - c)$,$(b, c - a)$,and $(c, a - b)$ are the vertices of a triangle,then where does the centroid of the triangle lie?

The vertices of $\Delta PQR$ are $P(2, 1)$,$Q(-2, 3)$,and $R(4, 5)$. Find the equation of the median through the vertex $R$.

The orthocentre of the triangle formed by the lines $x + y = 1$ and $xy = 0$ is