Exercise 9 Page 609

Let's consider the following right triangle.

The hypotenuse is the side lying opposite the right angle. We want to determine which of the given points always lies on the hypotenuse. Let's recall the definitions of the given points.

Name	Definition
Incenter	The incenter of a triangle is the point where the triangle's angle bisectors intersect.
Orthocenter	The orthocenter of a triangle is the point where the triangle's altitudes intersect.
Centroid	The centroid of a triangle is the point of intersection of the three medians of the triangle.
Circumcenter	The circumcenter of a triangle is the center of the circle circumscribing the triangle.

Let's determine where each of these points lies in a right triangle.

Since the incenter is the center of a circle inscribed in the triangle, it always lies inside of the triangle.

Note that the centroid will also lie inside of a right triangle.

Next we can see that the orthocenter of a right triangle is the vertex of the right angle. This is because the legs of a right triangle are two of its altitudes.

The circumcenter of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle. Thus, we have that the circumcenter of a right triangle lies in the middle of the hypotenuse. This corresponds to option D.