A pentagon is a five sided geometrical figure; if the pentagon fits exactly inside some other geometrical figure (such as a circle) then it can be said to be inscribed in that figure.
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assuming this is a regular pentagon (all five sides are equal length) the center is the intersection of the intersection of perpendicular bisectors of each side and should also be the center of the circle in which it is inscribed
The lines inscribed as directed will result in a "star" being formed inside, and a pentagon will be formed in the middle of the star. The pentagram will have been separated into 11 regions by inscribing it as directed.
It was named after a pentagon, it wasn't the 'Pentagon' that named the 'pentagon'.
Hexagon * * * * * A pentagon
It is an inscribed quadrilateral or cyclic quadrilateral.