Nine Point Circle Activity

Below is triangle ABC, with the following points constructed:
G: centroid H: orthocenter K: circumcenter
a, b, c: the midpoints of sides, respectively
m, n, p: the feet of the altitudes from the vertices, respectively
u, v, w: the midpoints from each vertex to the orthocenter

Drag the vertices A, B, C around and observe what happens to the line segment and the circle through the midpoints of the sides.

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Questions to think about:

--What do you notice about the line segment? How about the circle?
--Do any of the nine points ever overlap? Try this: Make angle BAC as close to 90° as possible, and make sides AB and AC as close to congruent as possible. How many of the nine points can you see? Does that make sense?
--Construct the circumcircle. Is there any relationships between the circumcircle and the nine-point circle? How is the incircle related to the nine-point circle?

Brian Bisignani, July 31, 2008, Created with GeoGebra