I was thinking about the curious case of a triangle whose sides are in arithmetic progression. For the case of a right angled triangle, I could only find the case of 3,4 and 5(the Pythagorian triple). But is there any way to find whether there is any other Pythagorian triple which are in arithmetic progression? other than this? if not, what is the underlying property of the triangle that make this choice of 3,4 and 5 a unique?
Also I would like to know what are the properties of a general triangle whose sides are in arithmetic progresssion. In particular can we provve that the relation $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ implies that $\sin A,\sin B,\sin C$ are in arithmetic progression?
Where $a,b,c$ are the sides and $A,B,C$ are angles opposite to each side respectively
Give the name $k$ for the common value
$$k=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
(in fact $k=2R$).
Then the criteria $b=\frac12(a+c)$ for $a,b,c$ being an AP is equivalent to the criteria $k\sin B=\frac12(k\sin A +k\sin C)$. (just simplify by $k$) for $\sin A,\sin B,\sin C$ to be themselves in A.P.
Edit :
I was wondering how triangles with sides in A.P. could be represented ; here is a way where we have chosen to take WLOG the mid-segment as unit, placed on the $x$-axis as line segment $BC$, i.e. where the sidelengths are ordered in the following way :
$$1-r \le 1 \le 1+r \ \text{with} \ 0<r<1$$
Vertex $A$ is one of the two intersections of circles $\frak{C}$$(B,1-r)$ and $\frak{C}$$(C,1+r)$ ; let us choose the one situated in the upper half plane.
A little calculation shows that the locus of point $A$ is the arc of ellipse with equation :
$$ x^2+\tfrac43y^2=1$$
situated in the second quadrant (red arc) in order to have a unicity of representation.
Please note that
position $A_1$ of point $A$ corresponds to the single possible case of a right triangle, which, not surprisingly "is" triangle $\frac34:1:\frac54$, which, with our normalization, is "the same as" triangle $3:4:5$:
extreme position $A=(0,\tfrac{\sqrt{3}}{2})$ corresponds to the case of an equilateral triangle : $1:1:1$.
extreme position $A=(-\frac32,0)$ corresponds to the case of a flat triangle $0:1:2$.