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octagon

In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an 8-sided polygon or 8-gon.
A ''regular octagon'' has Schläfli symbol 〔.〕 and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges.
==Properties of the general octagon==

The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°.
If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).〔Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", ''Forum Geometricorum'' 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html〕
The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.〔

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