A Geometric Analysis of the Pentagon Interior

The pentagon is, internally,  completely divided in every one of its segments in Phi ratio.

We will show triangle BEC similar to triangle BGA, similar to triangle AFG, and that each of these triangles is a golden mean triangle, so that each diagonal of the pentagon is divided in Extreme and Mean Ratio by all of the other diagonals.

"  " means angle!

1)   ABC =  BAD =  DAE =  EAC =  BEA =  BCA =  BDA =  ADC =  CDE =  DEB =  AEC =  DCB =  DCE =  CBE =  DBE, because they are all inscribed angles subtending chords of equal length, viz, the sides of the pentagon.
 FAG =  ABG by identity.
AG and  BGA common to both triangle AFG and triangle BGA.
2) Therefore triangle AFG similar to triangle BGA by angle-side-angle.

 ABG =  BEC by identity.
side AB = side CE
 ABG =  BCE as above.
3) Therefore triangle BEC similar to triangle BGA by angle-side-angle

So we know now that FG is to GA as GA is to AB as AB is to BE.
But this is the precise definition of division into Extreme and Mean Ratio. We can write this as:
4) FG/GA = GA/AB = AB/BE. So these 4 lines are in Phi ratio with each other.

Triangle BGA congruent to triangle CFA by angle-side-angle
Triangle BFA congruent to triangle AGC by angle-side-angle.
5) Therefore BF = FA = AG = GC and BC is divided into EMR by F and G

Triangle ADE congruent to triangle ADC congruent to triangle BEC congruent to triangle BCD
congruent to triangle BEA because all 3 angles are equal by 1) above and all of them have short sides equal to the side of the pentagon.
6) Therefore the lengths of the diagonals of the pentagon are equal to each other and so each of them is divided into EMR.

Further looks:
1–6  allows us to conclude that:
7) Triangles AFG, BFH, HDJ, EJI, and CGI are congruent.

8) Triangles  BGA, CFA, ABH, DBF, BDJ, EDH, DEI, CEJ, CEG and ACI are congruent.

All triangles in 7) and in 8) are golden mean triangles!

9) Since, for example, AB = BG, and  BC, a diagonal, is divided into EMR by BG,
THE DIAGONAL OF THE PENTAGON IS DIVIDED INTO EMR BY  THE SIDE OF THE PENTAGON.  This will become very useful in the analysis of certain polyhedra.

On to Pentagon Overview

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