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.

## Geometric Proof of the Regular Pentagon Construction

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- imatrendytotebag
**Posts:**152**Joined:**Thu Nov 29, 2007 1:16 am UTC

### Geometric Proof of the Regular Pentagon Construction

Hey baby, I'm proving love at nth sight by induction and you're my base case.

- BlackSails
**Posts:**5315**Joined:**Thu Dec 20, 2007 5:48 am UTC

### Re: Geometric Proof of the Regular Pentagon Construction

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.

- GenericPseudonym
**Posts:**206**Joined:**Thu Nov 11, 2010 12:30 am UTC**Location:**Wherever I happen to be.

### Re: Geometric Proof of the Regular Pentagon Construction

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.

- jestingrabbit
- Factoids are just Datas that haven't grown up yet
**Posts:**5967**Joined:**Tue Nov 28, 2006 9:50 pm UTC**Location:**Sydney

### Re: Geometric Proof of the Regular Pentagon Construction

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.

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.

ameretrifle wrote:Magic space feudalism is therefore a viable idea.

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