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In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle △ABC and a triple of angles α, β, γ. This information is sufficient to determine three points X, Y, Z such that $${\displaystyle {\begin{aligned}\angle ZAB&=\angle YAC&=\alpha ,\\\angle XBC&=\angle ZBA&=\beta ,\\\angle YCA&=\angle XCB&=\gamma .\end{aligned}}}$$ Then, by a theorem of Karl Friedrich Andreas Jacobi [de], the lines AX, BY, CZ are concurrent, at a point N called the Jacobi point.

The Jacobi point is a generalization of the Fermat point, which is obtained by letting α = β = γ = 60° and △ABC having no angle being greater or equal to 120°.

If the three angles above are equal, then N lies on the rectangular hyperbola given in areal coordinates by

$${\displaystyle yz(\cot B-\cot C)+zx(\cot C-\cot A)+xy(\cot A-\cot B)=0,}$$

which is Kiepert's hyperbola. Each choice of three equal angles determines a triangle center.

The Jacobi point can be further generalized as follows: If points K, L, M, N, O and P are constructed on the sides of triangle ABC so that BK/KC = CL/LB = CM/MA = AN/NC = AO/OB = BP/PA, triangles OPD, KLE and MNF are constructed so that ∠DOP = ∠FNM, ∠DPO = ∠EKL, ∠ELK = ∠FMN and triangles LMY, NOZ and PKX are respectively similar to triangles OPD, KLE and MNF, then DY, EZ and FX are concurrent.

References

1. de Villiers, Michael (2009). Some Adventures in Euclidean Geometry. Dynamic Mathematics Learning. pp. 138–140. ISBN 9780557102952.
2. Glenn T. Vickers, "Reciprocal Jacobi Triangles and the McCay Cubic", Forum Geometricorum 15, 2015, 179–183. http://forumgeom.fau.edu/FG2015volume15/FG201518.pdf Archived 2018-04-24 at the Wayback Machine
3. ^ Glenn T. Vickers, "The 19 Congruent Jacobi Triangles", Forum Geometricorum 16, 2016, 339–344. http://forumgeom.fau.edu/FG2016volume16/FG201642.pdf Archived 2018-04-24 at the Wayback Machine
4. Michael de Villiers, "A further generalization of the Fermat-Torricelli point", Mathematical Gazette, 1999, 14–16. https://www.researchgate.net/publication/270309612_8306_A_Further_Generalisation_of_the_Fermat-Torricelli_Point

External links

• A simple proof of Jacobi's theorem   written by Kostas Vittas
• Fermat-Torricelli generalization at Dynamic Geometry Sketches First interactive sketch generalizes the Fermat-Torricelli point to the Jacobi point, while 2nd one gives a further generalization of the Jacobi point.

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