Problem
GEO-B1-M06-P013 Six Equal Triangles
In triangle \(ABC\), medians \(AD\), \(BE\), \(CF\) meet at point \(G\). Prove that the six triangles \(AGF\), \(BGF\), \(BGD\), \(CGD\), \(CGE\), \(AGE\) have equal areas.
Use that each median halves the area, then compare the halves.
Denote the areas of \(AGF\), \(BGF\), \(BGD\), \(CGD\), \(CGE\), \(AGE\) in order by \(x_1,x_2,x_3,x_4,x_5,x_6\). Since \(F,D,E\) are side midpoints, we get \(x_1=x_2\), \(x_3=x_4\), \(x_5=x_6\). Median \(AD\) halves the whole triangle, so \(x_1+x_2+x_3=x_4+x_5+x_6\). Median \(BE\) gives \(x_1+x_2+x_6=x_3+x_4+x_5\). Substituting the pairwise equalities, we get \(x_1=x_5\) and \(x_1=x_3\). Hence all six areas are equal.
Can be discussed on the board by naming the areas; this is the first real area chasing problem.