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Napoleon's Triangle


Date: 08/10/1999 at 00:32:43
From: Eric Evenson
Subject: Napoleon's Triangle

What is Napoleon's triangle?


Date: 08/10/1999 at 04:17:44
From: Doctor Floor
Subject: Re: Napoleon's Triangle

Dear Eric,

Thanks for writing to Ask Dr. Math.

Let ABC be a given triangle. Construct the point D such that ABD is an 
equilateral triangle, and D and C are on opposite sides of AB. In the 
same way, find point E such that BCE is equilateral, and point F such 
that CAF is equilateral. Let J be the center of ABD, K the center of 
BCE, and L the center of CAF. Then JKL is Napoleon's triangle:


The interesting thing about Napoleon's triangle is that it is 
equilateral.

To see this, find point O such that triangles DAO and ABC are 
(directly) congruent. In the same way, find point N such that BDN and 
ABC are directly similar. Construct Napoleon triangles DAO and BDN: 


Now consider triangle TEB in the picture. Note that BE = BC and 
BT = BN = AC. Note also that:

     <EBT = 360 - 3*60 - <ABC - <DBN. 

Since angle DBN is equal to angle BAC, we find that 

     <EBT = 180 - <ABC - <BAC = <ACB. 

But that means that triangle TEB must be congruent to triangle ABC 
(SAS).

We can conclude that, by the way it is constructed, GK = KL, and using 
the same reasoning show that GK = KL = LM = MI = IH = HG [1].

Note that JL = JG by construction, GK = LK by [1], and of course 
JK = JK, so that triangles GJK and LJK are congruent (SSS). Thus 
<KJG = <LJK and we see that the six angles at J must all be congruent 
and are all 60 degrees. In particular, <LJK = 60 degrees.

In the same way we can show that <JKL and <KLJ are 60 degrees, and thus
triangle JKL is equilateral, as desired.

Another interesting fact about Napoleon's triangle is described in the 
Dr. Math archives:

   The Napoleon Point and More
    http://mathforum.org/dr.math/problems/schultess9.4.98.html   

I hope this is what you were looking for. If you have any other math 
questions, send them to Dr. Math.

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

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