The construction of a Nine-Point Circle

T. Barron

The Nine Point Circle, named for the nine constructed points on it, has several neat mathematical properties. Every triangle has a nine point circle which is connected to both it's inscribed circle, circumscribed circle, and Euler Line. It was first proven by French mathematicians Jean-Victor Poncelet and Charles Brianchon in 1821.

Constructing the triangle:

* Draw points A, B, and C.

* Draw intersecting straight lines from each of these points to draw a triangle.

* Construct the following segments:

* Segment m between points A and C.

* Segment n between points A and B.

* Segments o between points B and C.

Constructing circumcenter:

* Find the midpoint on m and call it point D.

* Find the midpoint of n and call it point E.

* Find the midpoint of o and call in point F.

* Construct a perpendicular line from line m through point D and call this line p.

* Construct a perpendicular line from line n through point E and call this line q.

* Construct a perpendicular line from line o through point F and call this line r.

* Find the intersection of these lines. This point, G, is called the circumcenter.

 

Constructing orthocenter:

* Hide lines p, q, and r.

* Construct a perpendicular line on o through point A. Name the perpendicular line r and label the point where o and r cross as point I.

* Construct a perpendicular line on n through point C. Name the perpendicular line s and label the point where n and s cross as point J.

* Construct a perpendicular line on m through point B. Name the perpendicular line t and label the point where m and t cross as point K.

* The lines constructed above are the altitudes of the triangle. The point of intersection, H, of the altitudes is called the orthocenter.

Finding the center, N, of the 9-point circle:

* Hide lines r, s, and t.

* Construct a segment between point G (the circumcenter) and point H (the orthocenter).

* Find the midpoint, N, of the this segment. This point, N, is the center of the Nine-Point Circle.

Finding the remaining 3 points:

* Construct segments from H to each of the points A, B, and C.

* Construct the midpoints of each of these lines and name them P, Q, and R, respectively.

The final construction of the circle:

* Construct a circle with the center N, picking one of the nine points as another point on the circle.

Click here to open a GSP Sketch of the Nine Point Circle or click here to open a GSP script to see how the Nine Point Circle was constructed. Also, you may click here to see a neat GSP animation of the Nine Point Circle.