Jacobi's theorem (geometry)

Adjacent colored angles are equal in measure. The point Template:Mvar is the Jacobi point for triangle Template:Math and these angles.

In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle Template:Math and a triple of angles Template:Mvar. This information is sufficient to determine three points Template:Mvar such that $$\begin{array}{ccc}\mathrm{\angle }ZAB& =\mathrm{\angle }YAC& =\alpha ,\\ \mathrm{\angle }XBC& =\mathrm{\angle }ZBA& =\beta ,\\ \mathrm{\angle }YCA& =\mathrm{\angle }XCB& =\gamma .\end{array}$$ Then, by a theorem of Template:Interlanguage link, the lines Template:Mvar are concurrent, at a point Template:Mvar called the Jacobi point.

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

If the three angles above are equal, then Template:Mvar 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.

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.

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