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 $${\displaystyle {\begin{aligned}\angle ZAB&=\angle YAC&=\alpha ,\\\angle XBC&=\angle ZBA&=\beta ,\\\angle YCA&=\angle XCB&=\gamma .\end{aligned}}}$$ Then, by a theorem of Karl Friedrich Andreas Jacobi [de], the lines AX, BY, CZ are concurrent,^[1][2][3] at a point N called the Jacobi point.^[3]

Adjacent colored angles are equal in measure. The point N is the Jacobi point for triangle △ABC and these angles.

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.^[4]