How do I prove that the triangle must be obtuse? Suppose you are given a triangle where the center of the nine-point circle lies on the circumcircle of the triangle. It is obvious that the triangle is obtuse, but how can you formally prove that the triangle must be obtuse?

Suppose that your triangle is acute. Then the orthocenter and circumcenter lie in the interior of the triangle. The nine-point center is the midpoint of these two points, hence lies in the interior of your triangle, by convexity.

Because the circumcircle intersects the triangle only at the three vertices, it is therefore impossible for the nine-point center to lie on the circumcircle.

This shows that such triangles may not be acute. In addition, they may not be right. To see why, recall that the nine-point center of a right triangle is given by the midpoint of the hypotenuse (which does not lie on the circumcircle).