Kuzas, Richmond, Tocchi – Team 6 – Nine Point Circle
Constructing The Nine-Point Circle of a Triangle
Goal: To construct the nine-point circle of a triangle and investigate its defining points.
1. Construct a triangle ABC .
[Triangle tool]
2. Create midpoints L, M, and N on segments AB , BC , and CA respectively.
[Midpoint tool]
3. Draw the altitudes of the vertices A, B, and C.
[Perpendicular & Segment tools]
4. Label the feet of the altitudes D, E, and F respectively.
[Label tool]
5. Create a point H where the three altitudes meet (the orthocenter).
[Point tool]
6. Hide lines AE , CD , and BF .
7. Draw segments AH , BH , and CH .
[Hide/Show tool]
[Segment tool]
8. Create and label the midpoints on each of the new segments X, Y, and Z respectively.
[Midpoint tool]
Kuzas, Richmond, Tocchi – Team 6 – Nine Point Circle
9. Use the circumcircle macro to locate the circumcircle for the ABC or find it
intersecting two perpendicular bisectors. Label the circumcenter O.
10. Hide the circumcircle. Draw segment HO .
[Hide/show and Segment tools]
(Recall: this segment is part of the Euler Line, which is the line passing through the
centroid, the orthocenter, and the circumcenter of a triangle.)
11. Draw and label the midpoint of the new segment point U.
[Midpoint tool]
12. Draw a circle with center point U and passing through L.
This is the nine-point circle.
[Circle tool]
13. Count and name the common points to the circle you created and the triangle . How many
are there?
9
Which points are those? D, L, M, E, F, N, X, Y, and Z.
Kuzas, Richmond, Tocchi – Team 6 – Nine Point Circle
14. Will these points be always in the intersection of the nine-point circle of any triangle and
the triangle? Yes
How can you test your answer to the previous question?
Try various triangles.
15. Define the Nine-Point circle using what you have learned in this activity.
The nine-point circle is a circle that can be constructed in any given triangle through a
specific set of nine points which include the midpoint of each side of the triangle, the foot
of each altitude, and the midpoint of the line segment from each vertex of the triangle to
the orthocenter.
16. Make a macro to construct the nine-point circle of a triangle. Make sure that the nine
points are included as final objects. Draw different triangles to test your macro.
17. Using the nine-point circle macro, investigate what, if anything, happens to the points on
the circle for equilateral, isosceles, and right triangles.
With an equilateral triangle, my “nine point circle” only used 6 points as they were only
three places where the triangle touched the sides of the triangle. With an isosceles
triangle, all nine points appeared. A right triangle’s “nine point circle” is only generated
by using 5 points since the circle stays outside the triangle around the right corner edge.
Kuzas, Richmond, Tocchi – Team 6 – Nine Point Circle
Extension:
1. Find what relation, if any, exists between the nine point circle and the circumcircle of a
given triangle.
The radius of the circumcircle is always twice as long as the radius of the nine point
circle. On an equilateral triangle, the center of a circumcircle and the center of the nine
point circle are the same.
Kuzas, Richmond, Tocchi – Team 6 – Nine Point Circle
Journal Activity
Nine point circle of a Triangle
1. List all definitions and properties that you have learned in this activity.
2. Can you think of any applications of this topic?
3. Can you relate this topic/concepts with other(s) previously studied? Explain your answer.