I claim it's not true, and I show it with a counterexample.
In the xy-plane, let the following points be labelled as such:
A = (0, 1)
B = (1, 1)
C = (-1, 0)
D = (0, 0)
E = (1, 0)
F = (0, -1)
Triangle ABE shares point A with triangle ACD.
Triangle ACD shares point D with triangle EDF.
Triangle ABE shares point E with triangle EDF.
No line can pass through all the interiors of the three triangles.