Construction of a Pentagon on the cube Proof of Part I: Pentagon is Equilateral Proof of Part II: Pentagon is in One Plane Proof of Part III: Pentagon is Equiangular Proof of Part IV: Side of Dodec Form Apotome Construction of the Dodecahedron

And, since the sides of the pentagon are equal (as proven in Proof I), and the magenta line equals the cyan line the angle of the pentagon with a green dot equals the angle of the pentagon with a light green dot.

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