Nine-Point Circle in the Complex Plane

Drag the sliders to explore triangles with vertices on the unit circle and their nine-point circles. Use the bookmark for right triangles, moving the slider.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Three points on the unit circle can be specified by their polar angles:
The vertices of triangle , then, are at the complex points
The midpoints (red) of the sides , , are simply
These three points define the nine-point circle, which has its center at
and radius . The formula for the nine-point circle is
The points (blue) for the intersections of the altitudes on the extended baselines {AB, AC, BC} are
The orthocenter (point O), which is the intersection of the extended altitudes of the triangle, is
The final three points (green) on the nine-point circle are the midpoints of the segments from the vertices to the orthocenter:
Thus we have a complete solution for the nine-point circle. From this discussion, it is apparent that there exist solutions for degenerate triangles where two or all of the points are the same, yielding a line segment or a point. To show the degenerate solutions, set two or all three of the sliders to the same value.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+