plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, a Jacobi point is a point in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

determined by a

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

''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 , the lines ''AX, BY,'' and ''CZ'' are

concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...

,Glenn T. Vickers, "Reciprocal Jacobi Triangles and the McCay Cubic", ''Forum Geometricorum 15'', 2015, 179–183. http://forumgeom.fau.edu/FG2015volume15/FG201518.pdfGlenn T. Vickers, "The 19 Congruent Jacobi Triangles", ''

Forum Geometricorum ''Forum Geometricorum: A Journal on Classical Euclidean Geometry'' is a peer-reviewed open-access academic journal that specializes in mathematical research papers on Euclidean geometry. It was founded in 2001, is published by Florida Atlantic U ...

'' 16, 2016, 339–344. http://forumgeom.fau.edu/FG2016volume16/FG201642.pdf at a point ''N'' called the Jacobi point. The Jacobi point is a generalization of the

Fermat point In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest ...

, 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 In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...

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 In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...

References

Theorems about triangles {{elementary-geometry-stub