I'm studying mass points and I stumbled upon this problem:
"Consider $\Delta ABC$ with points $P, Q, R$ on $AB, BC, AC$ with $AP:PB=BQ:QC = CR:RA$. Then the centroid of the triangle formed by $AQ, BR, CP$ coincides with the centroid of the original triangle $\Delta ABC$."
I was able to solve it using mass points. I'm wondering whether the converse is true and whether it's possible to prove it using mass points:
"Consider $\Delta ABC$ with points $P, Q, R$ on $AB, BC, AC$. If the centroid of the triangle formed by $AQ, BR, CP$ coincides with the centroid of the original triangle $\Delta ABC$, then $AP:PB=BQ:QC = CR:RA$."
Edit: There was a request for clarification. The triangle formed by $AQ, BR, CP$ is the triangle formed when the three cevians are drawn. If they do not all intersect at the same point, they form a triangle.
