Pythagorean triangle-perimeter a square and diameter of the inscribed circle a cube

 

 
Find a Pythagorean triangle with perimeter   p   a square and diameter of the inscribed circle a cube

Let the sides be

(m^2 - n^2) \,x,     2 \, m \, n \, x,     (m^2 + n^2) \,x

p \; = \; 2 \, m \, x \, (m + n) \; = \; r^2

Then     r \; = \;  n \, x \, (m - n)

the diameter   d   is to be a cube,   say   r^3 \,/ \,s^3

d \; = \; 2 \, n \, x \, (m - n) \; = \; r^3 \,/ \,s^3

From the two values of   x

x \; = \; r^2 \,/ \,(2 \, m \, (m+n))
x \; = \; r^3 \,/ \,(2 \, n \, s^3 \, (m-n))

we get   r   in terms of   m, n, s
r^2 \,/ \,(2 \, m \, (m+n)) \; = \; r^3 \,/ \,(2 \, n \, s^3 \, (m-n))

r \; = \; (m \, s^3 \, (n-m)) \,/ \,(n \, (m+n))

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

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

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s