The Steiner–Lehmus theorem, a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

in elementary geometry, was formulated by

C. L. Lehmus Daniel Christian Ludolph Lehmus (July 3, 1780 in Soest – January 18, 1863 in Berlin) was a German mathematician, who today is best remembered for the Steiner–Lehmus theorem, that was named after him. Lehmus was the grandson of the German poe ...

and subsequently proved by Jakob Steiner. It states: : ''Every

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 ...

with two angle bisectors of equal lengths is isosceles''. The theorem was first mentioned in 1840 in a letter by C. L. Lehmus to C. Sturm, in which he asked for a purely geometric proof. Sturm passed the request on to other mathematicians and Steiner was among the first to provide a solution. The theorem became a rather popular topic in elementary geometry ever since with a somewhat regular publication of articles on it.

Direct proofs

The Steiner–Lehmus theorem can be proved using elementary geometry by proving the

contrapositive In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statemen ...

statement: if a triangle is ''not'' isosceles, then it does ''not'' have two angle bisectors of equal length. There is some controversy over whether a "direct" proof is possible; allegedly "direct" proofs have been published, but not everyone agrees that these proofs are "direct." For example, there exist simple

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

ic expressions for angle bisectors in terms of the sides of the triangle. Equating two of these expressions and algebraically manipulating the equation results in a product of two factors which equal 0, but only one of them (''a'' − ''b'') can equal 0 and the other must be positive. Thus ''a'' = ''b''. But this may not be considered direct as one must first argue about why the other factor cannot be 0.

John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches o ...

Alleged impossibility of "direct" proof of Steiner–Lehmus theorem
/ref> has argued that there can be no "equality-chasing" proof because the theorem (stated algebraically) does not hold over an arbitrary

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, or even when negative

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s are allowed as parameters. A precise definition of a "direct proof" inside both classical and intuitionistic logic has been provided by Victor Pambuccian,. who proved, without presenting the direct proofs, that direct proofs must exist in both the classical logic and the intuitionistic logic setting.

Notes

References & further reading

* John Horton Conway, Alex Ryba: ''The Steiner-Lehmus Angle Bisector Theorem''. In: Mircea Pitici (Hrsg.): ''The Best Writing on Mathematics 2015''. Princeton University Press, 2016, , pp. 154–166 *Alexander Ostermann, Gerhard Wanner: ''Geometry by Its History.'' Springer, 2012, pp. 224–225 *David Beran: ''SSA and the Steiner-Lehmus Theorem''. The Mathematics Teacher, Vol. 85, No. 5 (May 1992), pp. 381–383
JSTOR
*C. F. Parry: ''A Variation on the Steiner-Lehmus Theme''. The Mathematical Gazette, Vol. 62, No. 420 (June 1978), pp. 89–94
JSTOR
*Mordechai Lewin: ''On the Steiner-Lehmus Theorem''. Mathematics Magazine, Vol. 47, No. 2 (March 1974), pp. 87–89
JSTOR
*S. Abu-Saymeh, M. Hajja, H. A. ShahAli:

Forum Geometricorum 8, 2008, pp. 131–140 *V. Pambuccian, H. Struve, R. Struve: ''The Steiner-Lehmus theorem and triangles with congruent medians are isosceles hold in weak geometries.'' Beitraege zur Algebra und Geometrie 57 (2016), no. 2, 483–497

External links

* *Paul Yiu
''Euclidean Geometry Notes''
Lectures Notes, Florida Atlantic University, pp. 16–17 *Torsten Sillke
''Steiner–Lehmus Theorem''
extensive compilation of proofs on a website of the University of Bielefeld {{DEFAULTSORT:Steiner-Lehmus theorem Euclidean geometry Theorems about special triangles