Heron triangles of the form (u^2 + 2w^2, u^2 + 4w^2, 2u^2 + 2w^2)

 
 
If   x \; = \; u^2   and   y \; = \; 2 \, w^2

x + y,   x + 2 \, y,   2 \, x + y   are the sides of a triangle with rational area   = \; (x+y) \: \sqrt { \,(2 \, x \, y) \,}

a \; = \; x \; + \; y \; = \; u^2 \; + \; 2 \, w^2
b \; = \; x \; + \; 2 \, y \; = \; u^2 \; + \; 4 \, w^2
c \; = \; 2 \, x \; + \; y \; = \; 2 \, u^2 \; + \; 2 \, w^2

s \; = \; (a + b + c)/2
= \; (x + y + x + 2 \,y + 2 \,x + y)/2 \; = \; 2 \, (x+y)

s \; - \; a \; = \; 2 \, (x+y) \; - \; (x + y) \; = \; x \; + \; y
s \; - \; b \; = \; 2 \, (x+y) \; - \; (x + 2 \,y) \; = \; x
s \; - \; c \; = \; 2 \, (x+y) \; - \; (2 \,x + y) \; = \; y

\sqrt { \,(s \,(s - a) \,(s - b) \,(s - c)) \,}
= \; \sqrt { \,(2 \,(x+y) \,(x + y) \,(x) \,(y)) \,}
= \; (x+y) \: \sqrt { \,(2 \, x \, y) \,}

s \; = \; (a + b + c)/2
= \; (u^2 + 2 \, w^2 + u^2 + 4 \, w^2 + 2 \, u^2 + 2 \, w^2)/2
= \; 2 \, (u^2 + 2 \, w^2)

s \; - \; a \; = \; 2 \, (u^2 + 2 \, w^2) \; - \; (u^2 + 2 \, w^2) \; = \; u^2 + 2 \, w^2
s \; - \; b \; = \; 2 \, (u^2 + 2 \, w^2) \; - \; (u^2 + 4 \, w^2) \; = \; u^2
s \; - \; c \; = \; 2 \, (u^2 + 2 \, w^2) \; - \; (2 \, u^2 + 2 \, w^2) \; = \; 2 \, w^2

\sqrt { \,(s \,(s - a) \,(s - b) \,(s - c)) \,}
= \; \sqrt { \,(2 \,(u^2 + 2 \,w^2) \,(u^2 + 2 \,w^2) \,(u^2) \,(2 \,w^2)) \,}
= \; 2 \, u \, w \, (u^2 + 2 \, w^2)

 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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