The invalid step in the proof is Step 4: "Since A" F = DF, we can rotate using center F to take A" to D."
This step is invalid because having equal distances does not necessarily imply that the two points A" and D are the same. The rotation in Step 4 assumes that A" F coincides with DF, but there is no guarantee that A", the image of A after translation and dilation, aligns perfectly with D.
In a valid proof, you would need to show that the corresponding angles are equal after the sequence of dilations, translations, and rotations. The given proof skips this crucial step and incorrectly concludes similarity based solely on the equality of distances, which is not sufficient to establish similarity.