<Cir-2025-792>, hint

Three circles of equal radius r meet at single point S while all  touch the circle of radius R

internally , then  S must be the center O of circle radius R, moreover R=2r.

Circles w' and w" meet at O(=S) and P and centers line O'O" pass through the midpoint of OP.  Triangles OT'T" (OT'=OT") and OO'O" are similar.

Hence T'T" pass through point P.