The applet illustrates a problem suggested by Bui Quang Tuan:

The fact is practically algebraic.

Let X, Y, Z be the midpoints of AB, AC, and BC, respectively. Introduce a = AY = YC, c = TX = XC, and b = CZ = ZB. Any of the numbers a, b, c may be negative if the direction of the corresponding segments does not agree with that of AB.

For T to be the midpoint of AB we should have 2a + 2c = 2b - 2c, or a + c = b - c. But this is exactly the condition for X to be the midpoint of YZ, which it is (because PO = OQ.)