Draw one of the medians, i.e. a line segment from one of the corners to the midpoint of the opposite side. Place the corner at the origin and let the median run along the x axis. Thus half the triangle is above the x axis and half is below. But is it really half?
The side that is cut by the x axis is cut in half; that's the definition of a median. Take half the length of this side and multiply by sin(θ) to get the height of the triangle above the x axis. Go down instead of up and the same formula gives the depth of the triangle below the x axis. Call this height h, and let the length of the median be b. Now the area of the two half triangles is ½bh. But what about the centroid?
Integrate y times the width of the triangle at height y. Whether we go up or down, the width starts at b and decreases linearly to 0. The upper integral (y > 0) and the lower integral (y < 0) cancel, and the result is 0. Thus the center of mass lies on the x axis.
Draw a second median and the same argument holds. The centroid lies on the point of intersection between the first and second median. Since the centroid is unique, the third median also contains the centroid. This is a round-about way of proving a counterintuitive result: The three medians of a triangle intersect at a point.
Using stiff cardboard or wood and a straight-edge, cut out an arbitrary triangle. Draw the medians and mark their intersection. The triangle should balance at this point.