Heron triangles & half-angle formulae

 
Prove that

If the sides of a triangle are in arithmetic progression if and only if the cotangents of its half-angles

\cot \, (A/2),    \cot \, (B/2),    \cot \, (C/2)

are also in arithmetic progression

 

Also,   if and only if     \tan \, (A/2) \, \tan \, (C/2) \; = \; 1/3

 

The sequence of the squares of the side lengths is in arithmetic progression if and only if

\cot \, A,    \cot \, B,    \cot \, C   is also an arithmetic progression.

 

Trigonometric proof using the Law of cotangents
Trigonometry/Solving triangles by half-angle formulae

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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