Nine-Point Circle
About This MicroSim
The nine-point circle (also called the Euler circle or Feuerbach circle) is one of the most elegant discoveries in triangle geometry. It passes through exactly nine significant points associated with any triangle.
The Nine Points
| Group | Points | Color |
|---|---|---|
| Midpoints of sides | 3 points | Blue |
| Feet of altitudes | 3 points | Red |
| Midpoints from vertices to orthocenter | 3 points | Orange |
Key Properties
The Nine-Point Center (N)
- Located exactly midway between the orthocenter (H) and circumcenter (O)
- Lies on the Euler line
The Nine-Point Radius
The radius of the nine-point circle is exactly half the circumradius:
\[r_9 = \frac{R}{2}\]
Historical Significance
This remarkable circle was discovered independently by several mathematicians in the early 19th century, including Feuerbach, Brianchon, and Poncelet.
Interactions
- Drag vertices to reshape the triangle
- Toggle buttons to show/hide each group of three points
- Observe all nine points remaining on the circle
Learning Objectives
- Identify the nine special points on the nine-point circle
- Understand that N is the midpoint of segment OH
- Recognize that the nine-point radius equals half the circumradius
- Appreciate this elegant geometric relationship
Bloom's Taxonomy Level
Analyzing and Evaluating