Copy of 9-Point Circle Action (Part 1)
For every triangle that exists, there is a very special circle that passes through 9 points. Some of these points lie on the triangle itself, and some do not. The applet below will informally illustrate the construction of a triangle's 9-point circle. Be sure to change the locations of the triangle's BIG WHITE VERTICES each time before re-sliding the slider. It would also be wise to alter the locations of these vertices after you've constructed this 9-point circle. Take your time with this applet! Study its dynamics very carefully. Answer the questions that follow.
Questions: 1) Where exactly is the center of a triangle's 9-point circle located? That is, how would you describe to somebody how to locate it? Be sure to use appropriate vocabulary terms and be sure to be specific in your response! 1A) The center of the 9 point circle is located in the midpoint between the circumcenter and orthocenter. I would first find the 2 points and then use a ruler to measure where the midpoint of the 2 points is. 2) Describe the points that are located on a triangle's 9-point circle. What exactly are these points? That is, how do these points relate to features of the triangle itself? 2A) The points are midpoints, the feet of the altitudes and the midpoint of the line from each vertex to the orthocenter. 3) Does the center of a triangle's 9-point circle always lie inside the triangle? 3A) No, it can be outside the triangle. 4) Is it ever possible for any 2 or more of these 9 points to overlap? That is, did you observe any cases where 2 (or more) pink points coincide (lie on top of each other)? If so, describe any possible conditions/features of the triangle for which this behavior occurred. 4A)I did not observe any of the 2 points overlapping. Be sure to use the tools of GeoGebra to help you provide answer(s) to this question!