Jacobi's theorem (geometry)

Jacobi's theorem, angles of the same color are equal

The set of Jacobi , named after Karl Friedrich Andreas Jacobi is a statement in elementary geometry on triangles or about a particular produced from triangles hexagon.

For any given triangle , three more triangles , and , are built over its sides , so that at the corner points of the , and there are two equally large angles, i.e. , and applies. Jacobi's theorem now states that the three lines , and intersect at a common point . ${\displaystyle ABC}$ ${\displaystyle ABD}$ ${\displaystyle BCE}$ ${\displaystyle ACF}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle \angle DAB=\angle CAF}$ ${\displaystyle \angle ABD=\angle CBE}$ ${\displaystyle \angle BCE=\angle ACF}$ ${\displaystyle AE}$ ${\displaystyle BF}$ ${\displaystyle CD}$ ${\displaystyle G}$

The common point of intersection is called the Jacobian point . Note that the Jacobi point is a property of the hexagon and not of the starting triangle, because in addition to the triangle it also depends on the angles at its three corner points. It can be understood as a generalization of the Fermat point , which is obtained when the starting triangle does not have an angle greater than and the angles at the corners of the triangle are or equilateral triangles are built over the sides of the triangle. ${\displaystyle G}$ ${\displaystyle ADBECF}$ ${\displaystyle ABC}$ ${\displaystyle ABC}$ ${\displaystyle 120^{\circ }}$ ${\displaystyle 60^{\circ }}$

Jacobi's theorem generalizes Kiepert's theorem , which considers the establishment of isosceles triangles with equal base angles across the sides of the triangle . ${\displaystyle ABC}$

literature

• Glenn T. Vickers: Reciprocal Jacobi Triangles and the McCay Cu . In: Forum Geometricorum , Volume 15, 2015, ISSN  1534-1178 , pp. 179-183
• Hans Walser: 99 Points of Intersection: Examples-Pictures-Proofs . MAA, 2006, ISBN 9780883855539 , p. 135

Web links

• Kiepert's And Jacobi's Theorems on cut-the-knot.org