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, by applying the same method to the other vertices of the pentagon, we obtain other similar straight lines. Since every straight line is in one plane, we proof that the pentagon is in one plane.

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