Triangles with sides in arithmetic progression (Q2012010)

It is shown that the sides \(a=BC\), \(b=AC\), \(c=AB\) of a triangle \(ABC\) are in arithmetic progression (\(2b=a+c\)) if and only if one of the following three conditions holds: (i) \(IG\) is parallel to \(AC\); (ii) \(OI \perp BI\); (iii) the midpoint of the segment joining \(B\) with the excenter (center of the excircle) opposite to \(B\) lies on \(AC\). Here, \(O\), \(I\), and \(G\) stand, as usual, for the circumcenter, the incenter, and the centroid.