Question

Call a relation R “orbital” if x R yand y R zimply z R x. Prove that R is an equivalence relation if and only R is both reflexive and orbital. (Note that this is an “if and only if” statement, which is biconditional. So there are actually two different implications to show here.)

This is what I did, but I don't think that it is the right approach:
If R is an equivalence relation then R has to be reflexive, symmetric, and transitive. This means that xRx which is reflexive. Also, xRy and yRx which means that it is symmetric.Then, R is also orbital or transitive if xRy and yRz which implies that zRx. This means that R is an equivalence relation. If R is reflexive then xRx. If R is orbital, then xRy and yRz which implies that xRz. This means that R is an equivalence relation.