I was going to ask this question but the problem has been posted here: Finding two points B and C such that the perimeter of triangle ABC is minimal . I tried to confirm the solution using the software and it does confirm that the triangle has the smallest perimeter. However, it feels like the answer(the proof) is incomplete. Consider the figure I generated below.
I simply choose a random points A,B on $OX$ and $OY$, and constructed the triangle. Then, I added the circles to generate the segment along the line $BC$.
I have satisfied the condition : $$AB+BC+AC=A_xB+BC+A_yC=A_xA_y$$ By simply leaving the argument at that then I also have a triangle whose perimeter is minimum. However, it is not. What makes the triangle Arnaldo made to be the correct triangle.