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 Karl Friedrich Andreas Jacobi, 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 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

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. 1 2 Glenn T. Vickers, "The 19 Congruent Jacobi Triangles", Forum Geometricorum 16, 2016, 339–344. http://forumgeom.fau.edu/FG2016volume16/FG201642.pdf
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