Nine-Point Circle in the Complex Plane

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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.

Contributed by: Bob Rimmer (January 2012)
Open content licensed under CC BY-NC-SA



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.

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