This question is from the book Abstract Algebra: Theory and Applications by Thomas W. Judson.
Chapter 1 – Question 21
Prove the relation defined on $\mathbb{R}^2$ by $(x_1, y_1) \sim (x_2, y_2)$ if $x_1^2 + y_1^2 = x_2^2 + y_2^2$ is an equivalence relation.