Jacobi's theorem (geometry)

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,^[1][2][3] at a point N called the Jacobi point.^[3]

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

${\displaystyle yz(\cot B-\cot C)+zx(\cot C-\cot A)+xy(\cot A-\cot B)=0,}$

which is Kiepert's hyperbola. Each choice of three equal angles determines a triangle center.

References

1. de Villiers, Michael (2009). Some Adventures in Euclidean Geometry. Dynamic Mathematics Learning. pp. 138–140. ISBN 9780557102952.
2. Glenn T. Vickers, "Reciprocal Jacobi Triangles and the McCay Cubic", Forum Geometricorum 15, 2015, 179–183. http://forumgeom.fau.edu/FG2015volume15/FG201518.pdf
3. Glenn T. Vickers, "The 19 Congruent Jacobi Triangles", Forum Geometricorum 16, 2016, 339–344. http://forumgeom.fau.edu/FG2016volume16/FG201642.pdf