Two Perspectors Are Collinear with the Incenter (Open problem)

Let ABC be a triangle. Let A', B', C' be the touch points of its incircle. Draw two internal semicircles with diameter BC', AC' & cyclic. Let semicircles BC', AC' meet the incircle in X, X', respectively. Define YY', ZZ' cyclically. Now the segments XX',YY', ZZ' form the triangle A_1, B_1, C_1. Also, let
B'' be the second intersection of semicircles BC', BA'. Define A'', C'' cyclically.
Then,

1) ABC is in perspective with A''B''C'' (in the incenter). This is trivial.
2) ABC is in perspective with A_1B_1C_1.
3) A''B''C'' is in perspective with A_1B_1C_1.
4) The three perspectors are collinear.

For more information see ADGEOM 1155