Show that a relation is equivalent if it is both reflexive and cyclic. A relation $R$ on set $X$ is called cyclic if whenever both $xRy$ and $yRz$ then $zRx$ where $x,y,z\in X$. Show that a relation on $X$ is an equivalence relation if and only if it is both reflexive and cyclic.

Since an equivalence relation is reflexive, symmetric and transitive and we already know the relation is reflexive I assume you prove that symmetric and transitive is the same as reflexive and cyclic. I honestly don't know where to start though or if my intuition is correct so any help would be great!
1) reflexive + cyclic => equivalence

a) reflexive => reflix

b) reflexive + cyclic => symmetry

xRy => xRy and yRy => yRx.

c) symmetry + cyclic => transitivity

xRy + yRz => zRx => xRz

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2)  Equivalence => reflixe + cyclic

a) reflexive => reflexive

b) transitive + symmetry => cyclic

xRy and yRz => xRz => zRx.
1) reflexive + cyclic => equivalence

a) reflexive => reflix

b) reflexive + cyclic => symmetry

xRy => xRy and yRy => yRx.

c) symmetry + cyclic => transitivity

xRy + yRz => zRx => xRz

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2)  Equivalence => reflixe + cyclic

a) reflexive => reflexive

b) transitive + symmetry => cyclic

xRy and yRz => xRz => zRx.