What would be enough or sufficient information to show that a certain point is a barycenter of a triangle?

Question

I posted a question on how to show that point was the barycenter of a triangle. I understood the process of the answer that they gave me, but don't know if the information is enough to show that point was the barycenterI since I don't know much about it. Here would be a drawing of what he basically did did:

He showed that $D$ was the midpoint of $BC$ and that $ \frac {BF}{BE}= \frac {2}{3}$. Would that be enough to show that $F$ is the barycenter?

Also if I have a point in a triangle the splits two lines (coming from two different vertices) into two segments with the proportion $1:2$ is that enough to say that that point is the barycenter?

Answer 1

Draw median $\overline{BE'}$ and let $G = \overline{BE'} \cap \overline{AD} $. Suppose that $E' \neq E$.

Then $BG/BE = BF/BE$, and thus $\triangle FBG \sim \triangle EBE'$. This implies $\angle BGF = \angle BE'E$, or that $\overline{GF} \parallel \overline{E'E}$, which is false since $\overline{GF}$ and $\overline{E'E}$ intersect at $A$.

Thus, $E' = E$, so $\overline{BE}$ is a median. Then $F = \overline{BE} \cap \overline{AD}$ is the barycenter.

Answer 2

The barycenter divides each median into two parts, one the double of the other. Hence if $D$ is the midpoint of $BC$ and $\frac{BF}{BE}=\frac {2}{3}$, then $F$ is the barycenter.

If instead you know that $FE/BF=FD/AF=1/2$, then triangles $BAF$ and $DEF$ are similar with $DE/AB=1/2$ and $DE\parallel AB$. It follows that triangles $ABC$ and $EDC$ are also similar and $DC/BC=EC/AC=1/2$. Hence $AD$ and $BE$ are medians and $F$ is the barycenter.