You can construct a circle that passes through a number of special points in a triangle. Connect the orthocenter to the circumcenter: this is “Euler’s Segment.” The Centroid lies on this segment (at a third of the distance from the circumcenter to the orthocenter). A number of centers and related points of a triangle are all closely related by Euler’s Segment and its Nine-Point Circle for any triangle. Create an Euler Segment and its related Nine-Point Circle, whose center is the midpoint of the Euler Segment.
You can watch a six-minute video of this segment and circle at:
www.khanacademy.org/math/geometry/triangle-properties/triangle_property_review/v/euler-line.
The video shows a hand-drawn figure, but you can drag your dynamic figure to explore the relationships more accurately and dynamically.
Are you amazed at the complex relationships that this figure has? How can a simple generic triangle have all these special points with such complex relationships? Could this result from the dependencies that are constructed when you define the different centers in your custom tools?