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I am looking for the definition of "a circle inscribed in a square" (along with any references for where the definition came from, if possible).

I am also looking for any information on what is and is not known directly from the definition.
For example, is it "assumed" by the definition or does it need to be proven that

$1)$ the circle is tangent to the square at the midpoints of the square?
$2)$ the diameter of the circle is equal to the sides of the square?
$3)$ if opposite midpoints of the opposite sides square are joined, then the circle is halved?

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    https://en.wikipedia.org/wiki/Inscribed_figure - your three properties can in many contexts be considered well-known, but actually deserve (simple) proofs if you want to be formal. All follows by symmetries – 2017-01-07

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