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I recently learned of the terms AP and GP, which mean Algebraic Progression and Geometric Progression, respectively. As I understand it: given a constant, $k$, and the sides of a triangle, $a$, $b$, and $c$,

the triangle is in AP if $$\left\{\begin{matrix}a+k=b\\b+k=c\end{matrix}\right\},$$

and the triangle is in GP if $$\left\{\begin{matrix}ka=b\\kb=c\end{matrix}\right\}$$

I am trying to understand why we prove that a triangle is in AP or GP.

Do the definitions of AP triangles and GP triangles allow us to prove anything non-trivial about the triangles?

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    If I replace "has a significant meaning" by "possesses a certain number of non-trivial properties", certainly not . See for example (http://math.stackexchange.com/q/1840910) where, even with a supplementary constraint, the GP condition does not bring something very interesting. – 2017-01-05
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    See Kepler triangle (https://en.wikipedia.org/wiki/Kepler_triangle) – 2017-01-06

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