A simple characterization of Student's distributions and normal distributions

Let X1:3≤X2:3≤X3:3 be the order statistics of the independent and identically distributed random variables X1, X2, and X3. Using the regressional property E[(aX2:3−X1:3)2|X2:3=x]=E[(X3:3−aX2:3)2|X2:3=x], we characterize some distributions, including Student's tν distributions (ν>2) and normal distributions.