Primitive Inverse Pythagorean Triplets

Define a primitive inverse Pythagorean triplet as any ordered triplet $(x,y,z)$ where $x,y,z\in\mathbb{Z}^+$ and $\gcd{(x,y,z)}=1$ such that $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$

Which of the following statements are true for any primitive inverse Pythagorean triplet $(x,y,z)$ where $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$?

• A: There are infinitely many primitive inverse Pythagorean triplets.
• B: Each of $\large\frac{xz}{y}$, $\large\frac{yz}{x}$, and $\large\frac{xy}{z}$ will be perfect squares.
• C: $(x,y,\sqrt{x^2+y^2})$ will form a Pythagorean triplet (in other words, $\sqrt{x^2+y^2}$ is an integer).

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