Jacobi's theorem (geometry)

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

{\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, 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

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.

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