A triangle has an obtuse angle at one of its vertices. Of the four classical centres — incentre, centroid, circumcentre and orthocentre — exactly two still lie inside the triangle. Which two are they?

The incentre is the meeting point of the internal angle bisectors, which all run into the interior, so it is always inside. The centroid is the average of the three vertices, so it is always inside as well. The circumcentre of an obtuse triangle falls on the far side of the longest side, and the orthocentre falls outside because two of the altitudes meet only when produced beyond the triangle. So exactly the incentre and the centroid remain inside.