Primitive isosceles Heronian triangle with an even side of the form 4n + 2

 
Any number of the form   4 \,n + 2   may be used as the even side of a primitive isosceles Heronian triangle by using sides

 
a \; = \; 2 \, n^2 \; + \; 2 \, n \; + \; 1
b \; = \; 2 \, n^2 \; + \; 2 \, n \; + \; 1
c \; = \; 4 \, n \; + \; 2
 

Let  P   be the perimeter:

P \; = \; 2 \,(2 \, n^2 + 2 \, n + 1) + 4 \, n + 2 \; = \; 4 \,(n + 1)^2

 
If   A   is the area of a triangle whose sides have lengths   a, \; b,   and   c   then

4 \, A \; = \; \sqrt { \,(a+b+c) \,(a+b-c) \,(a-b+c) \,(-a+b+c) \,}
 

Let’s compute the area of the triangle

  a + b + c \; = \; 4 \, (n + 1)^2
  a + b - c \; = \; 4 \, n^2
  a - b + c \; = \; 2 \,(2 \, n + 1)
-a + b + c \; = \; 2 \,(2 \, n + 1)
 

4 \, A \; = \; \sqrt { \,4 \, (n+1)^2  \: (4 \, n^2)  \: 4(2 \, n + 1)^2 \,}
4 \, A \; = \; 8 \, n \,(n + 1) \,(2 \, n + 1)
A \; = \; 2 \, n \,(n + 1) \,(2 \, n + 1)

we find that it’s an integer.

 
Here are the first few primitives:

 
HERON EVEN SIDE

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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