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### Origami Equilateral Triangle

```
Date: 04/26/2001 at 00:06:02
From: Oliver
Subject: Folding Construction: Equilateral Triangle

Hi,

I have spent quite a long time trying to figure out how to create an
equilateral triangle by folding; not using the sides of a piece of
paper and using nothing but your hands and, of course, your brain.

I have tried to make a hexagon from a piece of paper by folding,
because that would give you six equilateral triangles, but I have had
no luck in creating a hexagon from only folding.

I would appreciate your help and I am thankful you took the time to

Oliver
```

```
Date: 04/26/2001 at 12:36:34
From: Doctor Peterson
Subject: Re: Folding Construction: Equilateral Triangle

Hi, Oliver.

I'm not quite sure what you mean by "not using the sides"; does that
mean the edge of the paper can't form one of the sides of the
triangle, or that you can't assume the sides are parallel, or what? I
think what I've done probably is allowed, regardless of the
interpretation.

Origami books teach a standard way to make a 60-degree angle by
folding, which you can use to accomplish the task. If you have two
parallel lines and a perpendicular to them, form a third parallel
midway between the first two, and then fold the perpendicular AB at an
angle so that point A lies on the midline (A'), while the fold passes
through point B. Then the fold BC will be at the desired angle:

|
---B-----------------------------
|\  \
| \    \
|  \       \
|    \        \
|     \           \
---D------\--------------A'------
|       \            /
|        \          /
|          \      /
|           \    /
|            \ /
---A-------------C---------------
|

From the fact that A'B = AB, you can prove easily that ABA' is an
equilateral triangle, so that angle ABC measures 30 degrees.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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