Triangle-Preserving Curve Families

A family of curves $\Gamma_m$ is called triangle-preserving if for every triangle $ABC$, some curve in the family contains three points $P,Q,R$ with $\triangle PQR$ congruent to $\triangle ABC$. Determine, with proof, which of the following families are triangle-preserving.

1. $y=|mx|$ ($m\in\mathbb R$). 2. $x^2+y^2=m$ ($m>0$). 3. $y=x^m$ ($m\in\mathbb R$). 4. $y=mx^2$ ($m\in\mathbb R^{*}$). 5. $xy=m$ ($m\in\mathbb R^{*}$).