Jacobi's theorem (geometry)

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

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.

Weblinks

• A simple proof of Jacobi's theorem   written by Kostas Vittas