Classifying a Triangle!

In $\triangle ABC$ , suppose that the points $M,N$ lie on the line segment $BC$ , the point $P$ lies on the line segment $CA$ , and the point $Q$ lies on the line segment $AB$ , such that $MNPQ$ is a square. Suppose further that:

$\large{\dfrac{AM}{AN} = \dfrac{AC+\sqrt{2}AB}{AB+\sqrt{2}AC}}$

Then $\triangle ABC$ is definitely a: