The Fuhrmann circle is a geometric construct associated with a non‑degenerate triangle in Euclidean plane geometry.
Definition
For a given triangle $ABC$ let $I_{A}, I_{B}, I_{C}$ denote the excenters opposite the vertices $A, B, C$ respectively. Denote by
$$ M_{A}= \text{midpoint of } AI_{A},\qquad M_{B}= \text{midpoint of } BI_{B},\qquad M_{C}= \text{midpoint of } CI_{C}. $$
The points $M_{A}, M_{B}, M_{C}$ are concyclic; the unique circle passing through them is called the Fuhrmann circle of triangle $ABC$.
Properties
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Center and radius – The center of the Fuhrmann circle coincides with the nine‑point centre $N$ of triangle $ABC$, i.e. the midpoint of the segment joining the circumcenter $O$ and the orthocenter $H$. Its radius equals one half of the circumradius $R$ of $ABC$: $$ \operatorname{rad}(\text{Fuhrmann circle}) = \frac{R}{2}. $$
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Relation to the excentral triangle – The Fuhrmann circle of $ABC$ is the nine‑point circle of the excentral triangle $\triangle I_{A}I_{B}I_{C}$. Consequently it passes through the midpoints of the sides of the excentral triangle and the feet of its altitudes.
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Special points on the circle – In addition to the three midpoints $M_{A},M_{B},M_{C}$, the Fuhrmann circle contains the circumcenter $O$ of $ABC$ and the reflections of the orthocenter $H$ across the sides of $ABC$.
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Historical note – The circle is named after the German mathematician Wilhelm Fuhrmann (1838–1904), who investigated several loci related to triangle excenters in the late 19th century.
References
- H. S. M. Coxeter, Introduction to Geometry, Wiley, 1969 – discussion of excentral triangles and associated circles.
- J. L. Coolidge, The Geometry of the Triangle, Oxford University Press, 1931 – description of the Fuhrmann circle.
- R. A. Johnson, Advanced Euclidean Geometry, Dover Publications, 2007 – contains a proof of the concyclicity of $M_{A},M_{B},M_{C}$.
No further widely recognized meanings of the term “Fuhrmann circle” have been documented in the mathematical literature.