Inplane geometry, aJacobi point is a point in theEuclidean plane determined by atriangle△ABC and a triple of anglesα, β, γ. This information is sufficient to determine three pointsX, Y, Z such thatThen, by a theorem ofKarl Friedrich Andreas Jacobi [de], the linesAX, BY, CZ areconcurrent,^[1][2][3] at a pointN called the Jacobi point.^[3]

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

If the three angles above are equal, thenN lies on therectangular hyperbola given inareal coordinates by

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

which isKiepert's hyperbola. Each choice of three equal angles determines atriangle center.

The Jacobi point can be further generalized as follows:If pointsK,L,M,N,O andP are constructed on the sides of triangleABC so thatBK/KC = CL/LB = CM/MA = AN/NC = AO/OB = BP/PA, trianglesOPD,KLE andMNF are constructed so that ∠DOP = ∠FNM, ∠DPO = ∠EKL, ∠ELK = ∠FMN and trianglesLMY,NOZ andPKX are respectively similar to trianglesOPD,KLE andMNF, thenDY,EZ andFX are concurrent.^[4]

• A simple proof of Jacobi's theorem  written by Kostas Vittas
• Fermat-Torricelli generalization atDynamic 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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