= Solution
Suppose $R$ is a <reflexive relation> and a <Euclidean relation>. To prove it is a <symmetric relation>, assume $xRy$. Reflexivity gives $xRx$, so applying the Euclidean implication to $xRy$ and $xRx$ gives $yRx$.
To prove it is a <transitive relation>, assume $xRy$ and $yRz$. The symmetry just established gives $yRx$. Apply the Euclidean implication with common predecessor $y$ to $yRx$ and $yRz$; the conclusion is $xRz$. Thus all three requirements hold:
$$
\boxed{R\text{ reflexive and Euclidean}\Longrightarrow R\text{ an equivalence relation}.}
$$
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