We have $A, B$ and $C$ be three points in a plane.

Let

$A\equiv \left(0,1\right), B\equiv \left(1,0\right) \& C\equiv \left(-1,0\right)$

Let $P\left(x,y\right)$ be any variable point. Then we have

$P{A}^2+P{B}^2=2P{C}^2$

$\Rightarrow {\left[\sqrt{{\left(x-0\right)}^2+{\left(y-1\right)}^2}\right]}^2+{\left[\sqrt{{\left(x-1\right)}^2+{\left(y-0\right)}^2}\right]}^2=2{\left[\sqrt{{\left(x+1\right)}^2+{\left(y-0\right)}^2}\right]}^2$

$\Rightarrow x^2+{\left(y-1\right)}^2+{\left(x-1\right)}^2+y^2=2\left\{{\left(x+1\right)}^2+y^2\right\}$

$\Rightarrow 6x+2y=0$

It is a linear equation. Hence, locus of point $P\left(x,y\right)$ is a straight line.