One of my colleagues was teaching centroids today, and one of her students wondered if the 6 triangles formed by the medians all had the same area. Interesting. Neither of us had heard of this before, but it seemed plausible.
So I went into GeoGebra and made a triangle with draggable vertices, constructed its medians, found the areas of the six sub-triangles ... and they are all equal no matter what shape I make the big triangle. (GeoGebra file attached. You can drag points A, B and C.)
So it seems true, but can we prove it? ... On and off during the day, together we managed to prove it. Fun day.