The three medians of a triangle intersect at its centroid. The centroid is the triangle’s balance point, or center of gravity. (In other words, if you made the triangle out of cardboard, and put its centroid on your finger, it would balance.) On each median, the distance from the vertex to the centroid is twice as long as the distance from the centroid to the midpoint of the side opposite the vertex. That means that the centroid is exactly 1/3 of the way from the midpoint of the side to the vertex of the triangle. Take a look at the following figure.
If you’re from Missouri (the Show-Me State), you might want to actually see how a triangle balances on its centroid. Cut a triangle of any shape out of a fairly stiff piece of cardboard. Carefully find the midpoints of two of the sides, and then draw the two medians to those midpoints. The centroid is where these medians cross. (You can draw in the third median if you like, but you don’t need it to find the centroid.) Now, using something with a small, flat top such as an unsharpened pencil, the triangle will balance if you place the centroid right in the center of the pencil’s tip.