Concurrency of the Medians of Triangles and the Centroid of Triangles

In triangle \(ABC\), \(D\), \(E\) and \(F\) are the midpoints of \(AB\), \(BC\) and \(AC\), respectively. \(AE\) and \(BF\) are known as the medians of triangle \(ABC\). \(DG\) and \(FH\) are straight lines that pass through \(D\) and \(F\), respectively, and are parallel to \(AE\).

Triangle ABC with medians and parallel lines

1. Prove that \(BG = GE = EH = HC\).
2. Let \(O\) be the point of intersection of the medians \(AE\) and \(BF\). Hence or otherwise, prove that \(BO = 2OF\) and \(AO = 2OE\).
3. Use the fact that the result in (ii) is valid for the point of intersection of any two medians of triangle \(ABC\) to prove the concurrency (intersecting at one point) of the \(3\) medians of a triangle, where \(O\) is the point of intersection of the \(3\) medians.

Note: The point \(O\) is known as the centroid of triangle \(ABC\) and \(O\) is the common point that divides each of the \(3\) medians in the ratio \(2:1\), measured from its vertex of triangle \(ABC\).

There are many other interesting properties of the centroid of a triangle, which will not be presented here due to limited space. Readers may explore by themselves.