Golden Triangle: What is It?

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 
bicycles. Please help me find out what the Golden Triangle is. Thank 
you.

Robin Byerly

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

Hi, Robin,

Thanks for your question.

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/