Triangle (a,b,c) – inradius and circumradius — (Part 1)

 
 
Establish the following

For any triangle   (a, b, c)

a \, b \; + \; b \, c \; + \; c \, a \; = \; s^2 \; + \; 4 \, R \, r \; + \; r^2,     and
a \, b \, c \; = \; 4 \, R \, r \, s

where   R   is the circumradius,   r   the inradius and   s   the semiperimeter.
Then   a, b, c   are in arithmetic progression iff

s^2 \; + \; 9 r^2 \; = \; 18 \, R \, r
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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