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### Golden Triangle: What is It?

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Date: 09/19/1999 at 16:27:01
From: Robin Byerly
Subject: Golden Triangle

I'm doing a project for math where I have to answer questions using
the Internet. I have gone to many different Web sites trying to figure
out what the Golden Triangle is, but I cannot find it. It has to do
with math but so far all my results have something to do with
you.

Robin Byerly
```

```
Date: 09/20/1999 at 05:34:49
From: Doctor Floor
Subject: Re: Golden Triangle

Hi, Robin,

Suppose we have a triangle ABC, such that <A = <B = 72 degrees
(< means angle) and <C = 36 degrees. Such a triangle is known as the
Golden Triangle.

Let D be the point on BC, such that AD is the angle bisector of <A.
Then triangle ABD is again a Golden Triangle.

When we let lengths AB = AD = CD = x and BD = 1, then we find:

AB : BD  =  BC : AB
x : 1   = (x+1) : x

This can be rewritten to:

x^2 = x + 1
x^2 -x - 1 = 0

The two solutions for x are x = 1/2 +/- sqrt(5)/2. Since AB > BD, in
this case we must have x = 1/2 + sqrt(5)/2.

The number 1/2 + sqrt(5)/2 is known as the Golden Ratio, or Golden
Mean. So BC : AB is this famous ratio; that's why this triangle is
called a Golden Triangle.

For more about the Golden Ratio, see our FAQ:

Golden Ratio, Fibonacci Sequence
http://mathforum.org/dr.math/faq/faq.golden.ratio.html

As an example of the appearance of Golden Triangles: the outside
triangles of a pentagram are Golden Triangles.

When we attach to AC and BC two triangles that are congruent to
triangle ACD, we find a regular pentagon.

I hope this helped. If you need more, just write us back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Definitions
High School Fibonacci Sequence/Golden Ratio
High School Geometry
High School Triangles and Other Polygons
Middle School Definitions
Middle School Geometry
Middle School Triangles and Other Polygons

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