Jacobi's theorem (geometry) explained

In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle and a triple of angles . This information is sufficient to determine three points such that Then, by a theorem of, the lines are concurrent,^{[1] [2] [3]} at a point called the Jacobi point.^[3]

The Jacobi point is a generalization of the Fermat point, which is obtained by letting and having no angle being greater or equal to 120°.

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

$$yz(\backslash cot\; B\; -\; \backslash cot\; C)\; +\; zx(\backslash cot\; C\; -\; \backslash cot\; A)\; +\; xy(\backslash cot\; A\; -\; \backslash cot\; B)\; =\; 0,$$

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.^[4]

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.

Notes and References

1. Book: de Villiers, Michael . Some Adventures in Euclidean Geometry. 2009. Dynamic Mathematics Learning. 9780557102952. 138–140.
2. Glenn T. Vickers, "Reciprocal Jacobi Triangles and the McCay Cubic", Forum Geometricorum 15, 2015, 179–183. http://forumgeom.fau.edu/FG2015volume15/FG201518.pdf
3. Glenn T. Vickers, "The 19 Congruent Jacobi Triangles", Forum Geometricorum 16, 2016, 339–344. http://forumgeom.fau.edu/FG2016volume16/FG201642.pdf
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