Steiner-Lehmus theorem

{\ displaystyle | AE | = | BD |, \, \ alpha = \ beta, \, \ gamma = \ delta}

{\ displaystyle \ Rightarrow \ triangle ABC {\ text {is isosceles}}}

The Steiner-Lehmus theorem is a theorem of elementary geometry about triangles .

It was first formulated by Christian Ludolf Lehmus and then proven by Jakob Steiner .

If two bisectors are the same length in a triangle, it is isosceles .

The sentence was mentioned for the first time in 1840 in a letter from CL Lehmus to Charles-François Sturm , in which this Sturm asked for elementary geometric proof of the statement. Sturm spread the problem among other mathematicians and Jakob Steiner was one of the first to provide a proof. Since then, the sentence has become a popular subject in elementary geometry, on which numerous publications appeared in the following 160 years.

literature

John Horton Conway , Alex Ryba: The Steiner-Lehmus Angle Bisector Theorem . In: Mircea Pitici (Ed.): The Best Writing on Mathematics 2015 . Princeton University Press, 2016, ISBN 978-1-4008-7337-1 , pp. 154-166
Alexander Ostermann, Gerhard Wanner: Geometry by Its History. Springer, 2012, pp. 224–225
S. Abu-Saymeh, M. Hajja, HA ShahAli: Another Variation on the Steiner-Lehmus Theme . In: Forum Geometricorum , 8, 2008, pp. 131–140
David Beran: SSA and the Steiner-Lehmus Theorem . In: The Mathematics Teacher , Vol. 85, No. 5 (May 1992), pp. 381-383 ( JSTOR 27967647 )
CF Parry: A Variation on the Steiner-Lehmus Theme . In: The Mathematical Gazette , Vol. 62, No. 420 (June 1978), pp. 89-94 ( JSTOR 3617662 )
Mordechai Lewin: On the Steiner-Lehmus Theorem . In: Mathematics Magazine , Vol. 47, No. 2 (March 1974), pp. 87-89 ( JSTOR 2688873 )
V. Pambuccian, H. Struve, R. Struve: The Steiner-Lehmus theorem and triangles with congruent medians are isosceles hold in weak geometries . In: Contributions to Algebra and Geometry , Volume 57, 2016, No. 2, pp. 483–497

Web links

Florian Modler: Forgotten sentences on the triangle (part 2). In: Matroids Matheplanet, August 27, 2006
Eric W. Weisstein : Steiner-Lehmus theorem . In: MathWorld (English).
Paul Yiu: Euclidean Geometry Notes (PDF; 1.0 MB) Script, Florida Atlantic University, pp. 15–16 (English)
Torsten Sillke: Steiner-Lehmus theorem . Collection of various proofs for the Steiner-Lehmus Theorem

Individual evidence

↑ Harold Scott MacDonald Coxeter , Samuel L. Greitzer: Geometry Revisited . Random House, New York 1967, pp. 14–16 ( The Steiner – Lehmus Theorem )
↑ Diane and Roy Dowling: The Lasting Legacy of Ludolph Lehmus (PDF; 388 kB). In: Manitoba Math Links 2, 3, 2002, pp. 3-4.

[1] Harold Scott MacDonald Coxeter , Samuel L. Greitzer: Geometry Revisited . Random House, New York 1967, pp. 14–16 ( The Steiner – Lehmus Theorem )

[2] Diane and Roy Dowling: The Lasting Legacy of Ludolph Lehmus (PDF; 388 kB). In: Manitoba Math Links 2, 3, 2002, pp. 3-4.