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
Therefore, according to Euclid's Book VI Proposition 32:

For two triangles:

1) when 2 sides are proportional to 2 sides:
(magneta line:red line(on LEFT square)   =    red line:purple line (on RIGHT square))

and

2) they are placed at an angle such that 2 sides are parallel to 2 sides:
(the blue lines are parallel to each other, the light green lines are parallel to each other)

then

3) the remaining straight lines are in a straight line. (see green line).

Thus, we have proved our argument that the green line is a straight line.
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