Generating Primitive isosceles Heronian triangles from PPT

 
Pythagorean triples:

m, \; n   are positive integers, with   m \; > \; n.   The integers

a \; = \; m^2 \; - \; n^2
b \; = \; 2 \, m \, n
c \; = \; m^2 \; + \; n^2

form a Pythagorean triple.

Perimeter   P \; = \; m^2 + n^2 + 2 \, m \, n + m^2 - n^2 \; = \; 2 \, m^2 + 2 \, m \, n
Area   A \; = \; m \, n \, (m^2 - n^2)

 

(1)   (c, \; c, \; 2 \, a)   —->   m^2 + n^2, \; m^2 + n^2, \; 2 \,(m^2 - n^2)
(2)   (c, \; c, \; 2 \, b)   —->   m^2 + n^2, \; m^2 + n^2, \; 4 \, m \, n

It’s clear that these triples satisfy the triangle inequality.

 

(1)

Let’s compute the area of the triangle

  (m^2 + n^2) \; + \; (m^2 + n^2) \; + \; 2 \,(m^2 - n^2) \; = \; 4 \, m^2
  (m^2 + n^2) \; + \; (m^2 + n^2) \; - \; 2 \,(m^2 - n^2) \; = \; 4 \, n^2
  (m^2 + n^2) \; - \; (m^2 + n^2) \; + \; 2 \,(m^2 - n^2) \; = \; 2 \,(m^2 - n^2)
-(m^2 + n^2) \; + \; (m^2 + n^2) \; + \; 2 \,(m^2 - n^2) \; = \; 2 \,(m^2 - n^2)

4 \, A_1 \; = \; \sqrt { \, (4 \, m^2) \,(4 \, n^2)  \: 2 \,(m^2 - n^2)  \: 2 \,(m^2 - n^2) \, }
4 \, A_1 \; = \; \sqrt { \, (16 (m n)^2  \: 4(m^2 - n^2)^2 \, }
4 \, A_1 \; = \; 8 \, m \, n \, (m^2 - n^2)

A_1 \; = \; 2 \, m \, n \, (m^2 - n^2)

we find that it’s an integer.
 

(2)

Let’s compute the area of the triangle

  (m^2 + n^2) \; + \; (m^2 + n^2) \; + \; 4 \, m \, n \; = \; 2(m + n)^2
  (m^2 + n^2) \; + \; (m^2 + n^2) \; - \; 4 \, m \, n \; = \; 2(m - n)^2
  (m^2 + n^2) \; - \; (m^2 + n^2) \; + \; 4 \, m \, n \; = \; 4 \, m \, n
-(m^2 + n^2) \; + \; (m^2 + n^2) \; + \; 4 \, m \, n \; = \; 4 \, m \, n

4 \, A_2 \; = \; \sqrt { \, 2 \,(m + n)^2  \: 2(m - n)^2  \: (4 \, m \, n)  \: (4 \, m \, n) \, }
4 \, A_2 \; = \; \sqrt { \, 4 \,(m^2 - n^2)^2  \: (4 \, m \, n)  \: (4 \, m \, n) \, }
4 \, A_2 \; = \; 8 \, m \, n \, (m^2 - n^2)

A_2 \; = \; 2 \, m \, n \, (m^2 - n^2)

we find that it’s an integer.

PYTHA HERON 1
PYTHA HERON 2
PYTHA HERON 3
PYTHA HERON 4
PYTHA HERON 5

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

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