Steiner-Lehmus theorem

Steiner-Lehmus theorem

thumb|right|300px|">|AE|=|BD|,,alpha=eta,,gamma=delta Rightarrow riangle ABC ext{ is isosceles}The Steiner-Lehmus theorem, a theorem in elementary geometry, was formulated by C. L. Lehmus and subsequently proved by Jakob Steiner.

: Any triangle 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. C. Sturm passed the request on to other mathematicians and Jakob 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. [Coxeter, H. S. M. and Greitzer, S. L. "The Steiner-Lehmus Theorem." §1.5 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 14-16, 1967.] [ [http://www.umanitoba.ca/science/mathematics/undergrad_info/Issue6.pdf Diane and Roy Dowling: "The Lasting Legacy of Ludolph Lehmus", Manitoba Math Links -Volume II- Issue 3, Spring 2002] ]

Impossibility of a direct proof

The Steiner-Lehmus theorem can be proved using elementary geometry by proving the contrapositive statement.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."
John Conway [ [http://cs.nyu.edu/pipermail/fom/2004-August/008394.html Alleged impossibility of "direct" proof of Steiner-Lehmus theorem] ] has argued that there can be no "equality-chasing" proof because the theorem is false over an arbitrary field.However, until someone formulates a precise definition of what a "direct proof" is,there remains room for debate.

References

External links

*MathWorld|title=Steiner-Lehmus theorem|urlname=Steiner-LehmusTheorem
* [http://www.math.fau.edu/yiu/EuclideanGeometryNotes.pdf Euclidean Geometry Notes by Paul Yiu]
* [http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/steiner-lehmus Compilation of various proofs and notes on the Steiner-Lehmus theorem]


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