Summary: For a given triangle, the median triangle is a triangle constructed from the medians of the given triangle. The median triangle always exists in Euclidean geometry, and the existence can be shown by proving that the medians always satisfy the triangle inequality. The familiar high-school construction gives an alternate proof of its existence. It is not obvious that the median triangle always exists in hyperbolic geometry however. This note shows that it does indeed exist, but the note also shows that the familiar Euclidean geometry construction never yields the median triangle in hyperbolic geometry.