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 $N$. This information is sufficient to determine three points $&alpha;, &beta;, &gamma;$ such that $$\begin{align} \angle ZAB &= \angle YAC &= \alpha, \\ \angle XBC &= \angle ZBA &= \beta, \\ \angle YCA &= \angle XCB &= \gamma. \end{align}$$ Then, by a theorem of, the lines $X, Y, Z$ are concurrent, at a point $AX, BY, CZ$ called the Jacobi point.

The Jacobi point is a generalization of the Fermat point, which is obtained by letting $△ABC$ and $&alpha; = &beta; = &gamma; = 60°$ 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.