What is the premise of applying projective theorem?

I remember a classic saying that our math teacher encountered this kind of problem before.

To prove the product formula, we must prove the proportional formula; To prove the proportional formula, it is necessary to prove the similarity of triangles.

As you said in your question, to prove the square of DG =AG×GF, you must prove AG:DG=DG:GF. To prove that AG:DG=DG:GF, remove triangles and prove that triangles are similar. Junior high school is usually like this.