In a triangle, which one of these triples of lines need NOT pass through a single common point?

Ceva's theorem says AD, BE and CF are concurrent exactly when (BD/DC) × (CE/EA) × (AF/FB) = 1. For the medians every ratio is 1, so the product is 1 and they meet at the centroid. For the internal bisectors the ratios are c/b, a/c and b/a, whose product is again 1, giving the incentre. The altitudes always meet at the orthocentre, outside the triangle when it is obtuse. But the last triple gives (1/2) × (1/2) × (1/2) = 1/8, which is not 1, so those three cevians bound a small triangle instead of meeting at a point.