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, 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.

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.