Jacobi's theorem (geometry)

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,Glenn T. Vickers, "Reciprocal Jacobi Triangles and the McCay Cubic", Forum Geometricorum 15, 2015, 179–183. http://forumgeom.fau.edu/FG2015volume15/FG201518.pdf Glenn T. Vickers, "The 19 Congruent Jacobi Triangles", Forum Geometricorum 16, 2016, 339–344. http://forumgeom.fau.edu/FG2016volume16/FG201642.pdf at a point called the Jacobi point.

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

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.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

References

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.