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Centroids of Triangles with Vertices on the Unit Circle

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Marden's theorem [1], which Dan Kalman calls "the most marvelous theorem in mathematics," [2] states that given a triangle in the complex plane, there is a unique ellipse (the Steiner inellipse) that is tangent to the midpoint of each side of the triangle. Further, if the vertices of are the points , then the foci of are the critical points of the polynomial , and the centroid of is the root of .

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Frayer et al. [3] note that if one of the vertices is fixed (say ), then as the other two vary, stays within a circle of radius whose center is of the way along a radial line through . This is a consequence of a fundamental theorem about triangle medians [4] that states the centroid divides the median of a triangle in a ratio. Let be the midpoint of the side opposite ; this means that . Since must be within the unit disk, it follows that .

In the image, the fixed vertex is red and the dashed blue circle contains the centroid , shown in green.

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Contributed by: Chris Boucher (August 2022)
Open content licensed under CC BY-NC-SA

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References

[1] B. Torrence, "Marden's Theorem" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/MardensTheorem.

[2] D. Kalman, "An Elementary Proof of Marden’s Theorem," The American Mathematical Monthly, 115(4), 2008 pp. 330–338. www.jstor.org/stable/27642475.

[3] C. Frayer, M. Kwon, C. Schafhauser and J. A. Swenson, "The Geometry of Cubic Polynomials," Mathematics Magazine, 87(2), 2014 pp. 113–124. doi:10.4169/math.mag.87.2.113.

[4] T. Garza, "The Centroid of a Triangle Divides Each Median in the Ratio 1:2" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/TheCentroidOfATriangleDividesEachMedianInTheRatio12.

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