## Pythagorean triples (a,b,c), (x,y,z) and (z+c)^2 – (x+a)^2 – (y+b)^2

The integers

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

form a Pythagorean triple

So do these integers

$x \; = \; u^2 \; - \; v^2$
$y \; = \; 2 \, u \, v$
$z \; = \; u^2 \; + \; v^2$

Consider

$(z + c)^2 = m^4+2 m^2 n^2+2 m^2 u^2+2 m^2 v^2+n^4+2 n^2 u^2+2 n^2 v^2+u^4+2 u^2 v^2+v^4$
$(x + a)^2 = m^4-2 m^2 n^2+2 m^2 u^2-2 m^2 v^2+n^4-2 n^2 u^2+2 n^2 v^2+u^4-2 u^2 v^2+v^4$
$(y + b)^2 = 4 m^2 n^2+8 m n u v+4 u^2 v^2$

The expression

$(z + c)^2 \; - \; (x + a)^2 \; - \; (y + b)^2$

gives us

$2 m^2 u^2+2 m^2 v^2+2 n^2 u^2+2 n^2 v^2 - (2 m^2 u^2-2 m^2 v^2-2 n^2 u^2+2 n^2 v^2) - 8 m n u v$

$= \; 4 m^2 v^2-8 m n u v+4 n^2 u^2$

$= \; 4 \, (n u-m v)^2$

The expression    $(z + c)^2 \; - \; (x + a)^2 \; - \; (y + b)^2 \; = \; 4 \, (n \, u - m \, v)^2$   is a square

Expression #2 :    $c \,z \; - \; b \,x \; - \; a \,y$

$= \; (m^2 + n^2) (u^2 + v^2) - 2 \, m \, n \, (u^2 - v^2) - 2 \, u \, v \, (m^2 - n^2)$
$= \; m^2 u^2-2 m^2 u v+m^2 v^2-2 m n u^2+2 m n v^2+n^2 u^2+2 n^2 u v+n^2 v^2$
$= \; (m \, u - m \, v - n \, u - n \, v)^2$

$c \,z - b \,x - a \,y \; = \; (m \, u - m \, v - n \, u - n \, v)^2$

Expression #3 :    $c \,z \; + \; b \,x \; + \; a \,y$

$= \; (m^2 + n^2) (u^2 + v^2) + 2 \, m \, n \, (u^2 - v^2) + 2 \, u \, v \, (m^2 - n^2)$
$= \; m^2 u^2+2 m^2 u v+m^2 v^2+2 m n u^2-2 m n v^2+n^2 u^2-2 n^2 u v+n^2 v^2$
$= \; (m \, u + m \, v + n \, u - n \, v)^2$

$c \,z \; + \; b \,x \; + \; a \,y \; = \; (m \, u + m \, v + n \, u - n \, v)^2$

Expression #4 :    $c \,z \; + \; b \,x \; - \; a \,y$

$= \; (m^2 + n^2) (u^2 + v^2) + 2 \, m \, n \, (u^2 - v^2) - 2 \, u \, v \, (m^2 - n^2)$
$= \; m^2 u^2-2 m^2 u v+m^2 v^2+2 m n u^2-2 m n v^2+n^2 u^2+2 n^2 u v+n^2 v^2$
$= \; (m \, u - m \, v + n \, u + n \, v)^2$

$c \,z \; + \; b \,x \; - \; a \,y \; = \; (m \, u - m \, v + n \, u + n \, v)^2$

Expression #5 :    $c \,z \; - \; b \,x \; + \; a \,y$

$= \; (m^2 + n^2) (u^2 + v^2) - 2 \, m \, n \, (u^2 - v^2) + 2 \, u \, v \, (m^2 - n^2)$
$= \; m^2 u^2+2 m^2 u v+m^2 v^2-2 m n u^2+2 m n v^2+n^2 u^2-2 n^2 u v+n^2 v^2$
$= \; (m \, u + m \, v - n \, u + n \, v)^2$

$c \,z \; - \; b \,x \; + \; a \,y \; = \; (m \, u + m \, v - n \, u + n \, v)^2$

In summary #1:

#1    $(z + c)^2 \; - \; (x + a)^2 \; - \; (y + b)^2 \; = \; 4 \, (n \, u - m \, v)^2$

#2    $c \,z \; - \; b \,x \; - \; a \,y \; = \; (m \, u \; - \; m \, v \; - \; n \, u \; - \; n \, v)^2$
#3    $c \,z \; + \; b \,x \; + \; a \,y \; = \; (m \, u \; + \; m \, v \; + \; n \, u \; - \; n \, v)^2$
#4    $c \,z \; + \; b \,x \; - \; a \,y \; = \; (m \, u \; - \; m \, v \; + \; n \, u \; + \; n \, v)^2$
#5    $c \,z \; - \; b \,x \; + \; a \,y \; = \; (m \, u \; + \; m \, v \; - \; n \, u \; + \; n \, v)^2$

Expression #6 :    $c \,z \; - \; a \,x \; - \; b \,y$

$= (m^2 + n^2) \,(u^2 + v^2) - (m^2 - n^2) \,(u^2 - v^2) - 4 \, m \, n \, u \, v$
$= 2 \, m^2 \, v^2 - 4 \, m \, n \, u \, v + 2 \, n^2 \, u^2$
$= 2 \, (n \, u - m \, v)^2$

$c \,z \; - \; a \,x \; - \; b \,y \; = \; 2 \, (n \, u - m \, v)^2$

Expression #7 :    $c \,z \; + \; a \,x \; + \; b \,y$

$= (m^2 + n^2) \,(u^2 + v^2) + (m^2 - n^2) \,(u^2 - v^2) + 4 \, m \, n \, u \, v$
$= 2 \, m^2 \, u^2 + 4 \, m \, n \, u \, v + 2 \, n^2 \, v^2$
$= 2 \, (m \, u + n \, v)^2$

$c \,z \; + \; a \,x \; + \; b \,y \; = \; 2 \, (m \, u + n \, v)^2$

Expression #8 :    $c \,z \; - \; a \,x \; + \; b \,y$

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

$c \,z \; - \; a \,x \; + \; b \,y \; = \; 2 \, (m \, v + n \, u)^2$

Expression #9 :    $c \,z \; + \; a \,x \; - \; b \,y$

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

$c \,z \; + \; a \,x \; - \; b \,y \; = \; 2 \,(m \, u - n \, v)^2$

In summary #2 :

#6    $c \,z \; - \; a \,x \; - \; b \,y \; = \; 2 \,(n \, u - m \, v)^2$
#7    $c \,z \; + \; a \,x \; + \; b \,y \; = \; 2 \,(m \, u + n \, v)^2$
#8    $c \,z \; - \; a \,x \; + \; b \,y \; = \; 2 \,(m \, v + n \, u)^2$
#9    $c \,z \; + \; a \,x \; - \; b \,y \; = \; 2 \,(m \, u - n \, v)^2$