Consider two points in the Euclidean plane:
and some fixed real number $\lambda\in(0,1)$. The claim is that the following expression is always a positive real number:
$$(1-\lambda^2)\cdot (\vert B \vert^2- \vert A\vert^2)+(A_1+B_1\cdot \lambda^2)^2+(A_2+B_2\cdot \lambda^2)^2>0.$$
I'm wondering why this complicated looking real number is always positive? I plugged in some numbers and it was positive. Is it possible to get the minimum of this function? Best regards.