Jacobi point

Adjacent colored angles are equal in measure. The point N is the Jacobi point for this triangle 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 P, Q, and R such that ∠RAB = ∠QAC = α, ∠PBC = ∠RBA = β, and ∠QCA = ∠PCB = γ. Then, by a theorem of Karl Friedrich Andreas Jacobi, the lines AP, BQ, and CR are concurrent, at a point N called the Jacobi point.

The Jacobi point is a generalization of the Napoleon points, which are obtained by letting α = β = γ = 60°.

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.

References

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