I haven't found a proof for this property (well, it's more of a guess), but I haven't found any counterexamples. I wanted to ask for help with either. \
Given a tetrahedron $ABCD$ where $AB = AC = AD$ (i.e these edges are of equal length), show that the line perpendicular to the triangle $BCD$ passing through the centroid of $BCD$ passes through $A$?
Is this true? (I am thinking about this in context of a different problem - the above statement is just my guess about the geometry of a tetrahedron). I'd appreciate either a reference to a proof or counterexample as I don't know how to go about this.