A student claims to have drawn a triangle in which one median is exactly half the length of the side it is drawn to, and which is nevertheless not right-angled. Why can no such triangle exist?

Let the median from A go to the midpoint M of BC, and suppose AM = ½BC. Then AM = BM = CM, so A is the same distance from M as B and C are, which puts A on the circle with centre M and radius ½BC — the circle having BC as a diameter. An angle in a semicircle is a right angle, so ∠BAC = 90° and the triangle is right-angled after all. The first option is false the other way round: the median to the hypotenuse of a right triangle is exactly half of it, and in an obtuse triangle the median to the longest side is shorter still. Nothing forces the median to be the longest segment, and the three medians of every triangle are concurrent at the centroid without exception.