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, and γ. This information is sufficient to determine three points X, Y, and Z such that ∠ZAB = ∠YAC = α, ∠XBC = ∠ZBA = β, and ∠YCA = ∠XCB = γ. Then, by a theorem of  [de], the lines AX, BY, and CZ are concurrent,^[1][2][3] at a point N called the Jacobi point.^[3]

The Jacobi point is a generalization of the Fermat point, which is obtained by letting α = β = γ = 60° and triangle 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.

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