Geometric Proof of the Regular Pentagon Construction

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Geometric Proof of the Regular Pentagon Construction

Postby imatrendytotebag » Thu Nov 04, 2010 4:18 am UTC

Hey all.

I've been wondering for some time if there was a construction of a regular pentagon that had a geometric proof that the construction really worked. The word "geometric" is fairly loose, but vaguely I mean using arguments like congruence/similarity of triangles, and angle arithmetic. It's not that the existing proof is "non-geometric", or that I have any qualms about it's validity, but it would be nice to see a way of approaching the construction that didn't involve first the calculation of cos(36).

Interested to see what you guys come up with.
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Re: Geometric Proof of the Regular Pentagon Construction

Postby Tirian » Thu Nov 04, 2010 3:55 pm UTC

I don't know if there are ways that conceal it better than others, but every straightedge and compass construction of the pentagon is implicitly about showing that cos 72 is a constructable number. You can't do the one without the other.

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Re: Geometric Proof of the Regular Pentagon Construction

Postby GenericPseudonym » Fri Nov 12, 2010 6:55 am UTC

How about this: there's some law somewhere that says the sum of the angles in a regular shape = 180(#sides - 2), which means for the pentagon it'll be 540. 540/5=108, so 108 degree angles. If you look at the pentagon like 5 isosceles triangles, the base angles of each triangle must be 54 degrees, which means the other angle is 180-54-54=72. Since 72*5=360, this is obviously a working pentagon. All you need to do to construct it is to draw 5 congruent 72-54-54 triangles which share legs, using the ol' compass and protractor. And since all the triangles'll be congruent, you know it's regular.

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Re: Geometric Proof of the Regular Pentagon Construction

Postby jestingrabbit » Fri Nov 12, 2010 7:17 am UTC

Whilst that would work, its not what was asked for. Usually, when we talk about a construction of a geometric figure, we mean a construction that only uses a straightedge and compass: no measuring angles, no measuring lengths.

When you work like this there are only some angles and lengths that are constructible, so only some polygons are constructible. You can read up on this idea here.
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