Let us have a triangle ABC in the Cartesian plane and consider the following transformation of this triangle:
On the ray AB starting at A, select a point B' so that so that |AB'|=|AC|. Likewise, on the ray BC starting at B, select a point C' so that BC' so that |BC'|=|AB|, and on the ray CA starting at C select a point A' so that |BA'|=|BC|. As a result, a new triangle A'B'C' is obtained.
Prove (or disprove) that the perimeter of a new triangle A'B'C' does not exceed the perimeter of ABC.
Remark. The question has been stated by Ahmet Yaşar Özban, Atilim University in regard with his research on iterative methods.