Nine Points and You Are OUT!
By Stephanie Britt
The nine-point circle occurs when certain constructions are done to a triangle.
According to definition the Nine-Point circle for any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and the three mid-points of the segments from the respective vertices to orthocenter.
The first triangle constructed is a non equilateral acute triangle.
The center of the circle was found by finding the perpindicular bisectors of the inner triangle and using their intersection. As shown below.
A good exploration is testing to see if the nine-point circle works for all types of triangles.
A right triangle:
As the right angle B approaches 90 degrees we can see some of the points on the circle converging with other points.
Once we move angle B into place, points 5, 6, and 7 converge to one, 3 and 9 become one and 2 and 8 combine.
Our nine-point circle has been reduced to a five-point circle with one of the points being a vertex of the triangle.
An obtuse triangle:
As we move angle A out to make angle B greater than 90 degrees we see that the center of the circle N moves closer to B and points 1 and 4 get closer together.
As angle B gets bigger B will become center N and points 1 and 4 will become one and create a three-point circle.
The only other type of triangle is an equilateral triangle:
We end up with a six-point circle.
With this exploration we can conclude that the nine-point circle only occurs within an acute non-equilateral triangle.