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?
The center of a triangle's 9-point circle is located exactly in the middle between the circumcenter and the orthocenter.
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?
For this triangle specifically, three of the points that lie on the 9-point circle are the midpoints of each side of the triangle itself. Another three points that lie on the 9-point circle are where the altitudes are formed. Finally, the last three points on the circle are formed when the orthocenter was connected to each vertex and then the midpoint was found of each of those lines.
3) Does the center of a triangle's 9-point circle always lie inside the triangle?
The center of a triangle's 9-point circle does not always lie inside the triangle, it depends on the type of triangle. For example, if the triangle is a right or acute triangle the center of the 9-point circle will lie inside the triangle, but if it is an obtuse triangle the center will lie outside of 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.
It is possible for 2 or more of these 9 points to overlap. For example, in a right triangle, three of the points overlapped on the vertex of the right angle.