Primitive Inverse Pythagorean Triplets

Define a primitive inverse Pythagorean triplet as any ordered triplet (x,y,z)(x,y,z) where x,y,zZ+x,y,z\in\mathbb{Z}^+ and gcd(x,y,z)=1\gcd{(x,y,z)}=1 such that 1x2+1y2=1z2\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)(x,y,z) where 1x2+1y2=1z2\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}?

  • A: There are infinitely many primitive inverse Pythagorean triplets.
  • B: Each of xzy\large\frac{xz}{y}, yzx\large\frac{yz}{x}, and xyz\large\frac{xy}{z} will be perfect squares.
  • C: (x,y,x2+y2)(x,y,\sqrt{x^2+y^2}) will form a Pythagorean triplet (in other words, x2+y2\sqrt{x^2+y^2} is an integer).

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