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 in this picture, in the right triangle closest to us: square of cyan line  +  square of green line =  square of black line. But, in the previous page, we proved that: square of cyan line  +  square of green line = 4 x square of green line. Therefore: square of black line  =  4 x square of green line and thus: black line  =  2 x the green line. Since the red line  =  2 x the green line, the red line  =  the black line. Therefore, any two sides of the pentagon, constructed in this manner, will be equal, and the Constructed Pentagon is Equilateral.

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