Points A, B, C can be moved; number t can be varied between 0 and 1 by moving its dot along a slider.

Point D is the result of moving A toward B by factor t, i.e., distance(A,D) = t ∗ distance(A,B); points E and F are obtained in a similar fashion by moving B towards C and C towards A using the same factor. (That operation is called a dilation.)

Are there any non-trivial choices of t such that triangle DEF is similar to triangle ABC? (Note: if t=0 or t=1, then the two triangles are congruent.)