Show that the ninepoint circle of triangle ABC is the locus

Show that the nine-point circle of triangle ABC is the locus of all midpoints of segments UH, where H is the orthocenter of Triangle ABC and U is an arbitrary point of the circumcircle. HINT: We already know that this locus is a circle.

Solution

By the definition of a nine-point circle, the circle passes through nine significant points:

The circumcircle passes through all the three vertices of the triangle. So, we already know that the three significant points are the midpoints of UH where H is the orthocenter and U is a point on the circumcircle.

We have a property for the nine point circle for a triangle that a nine-point circle bisects a line segment going from the corresponding triangle\'s orthocenter to any point on its circumcircle.

Since this is true for any point on the circumcircle and we already know that the locus of the midpoints of the segments joining the orthocenter and a point on the circumcircle is a circle, the result follows that the locus circle is the nin-point circle.