Proving the infimum of a function of four variables is positive

Question

I've got this function $$f : \{(x_1,x_2,y_1,y_2)\in\Bbb{R}^4\mid x_1\neq x_2 \text{ or }y_1\neq y_2\}\to \Bbb R$$ defined by $$f(x_1,x_2,y_1,y_2) = \frac{(x_1^2+x_2^2)^2+3(y_1^2+y_2^2)^2-4(y_1^2+y_2^2)(x_1y_1+x_2y_2)}{(y_1^2+y_2^2)^2+3(x_1^2+x_2^2)^2-4(x_1^2+x_2^2)(x_1y_1+x_2y_2)}$$ that I'd like to show that the infimum is positive. I'm not entirely sure where to start, as I've tried to doing some basic algebra alterations to the numerator and denominator but that hasn't gotten me anywhere.