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The Steiner-Lehmus angle-bisector theorem

Published online by Cambridge University Press:  23 January 2015

John Conway  and
Alex Ryba
John Conway
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
Alex Ryba
Affiliation:
Department of Computer Science, Queens CollegeCUNY, 65-30 Kissena Blvd, Flushing NY 11367, USA, email:ryba@sylow.cs.qc.cuny.edu
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Extract

In 1840 C. L. Lehmus sent the following problem to Charles Sturm: ‘If two angle bisectors of a triangle have equal length, is the triangle necessarily isosceles?’ The answer is ‘yes’, and indeed we have the reverse-comparison theorem: Of two unequal angles, the larger has the shorter bisector (see [1, 2]).

Sturm passed the problem on to other mathematicians, in particular to the great Swiss geometer Jakob Steiner, who provided a proof. In this paper we give several proofs and discuss the old query: ‘Is there a direct proof?’ before suggesting that this is no longer the right question to ask.

We go on to discuss all cases when an angle bisector (internal orexternal) of some angle is equal to one of another.

Type
Articles
Information
The Mathematical Gazette , Volume 98 , Issue 542 , July 2014 , pp. 193 - 203
Copyright
Copyright © Mathematical Association 2014 

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References

1. Coxeter, H. S. M. and Greitzer, S. L., Geometry Revisited, Mathematical Association of America (1967).Google Scholar
2. Leversha, G., The geometry of the triangle, UKMT (2013).Google Scholar
3. Gardner, Sherri R., A variety of proofs of the Steiner-Lehmus theorem, East Tennessee State University, Master's thesis (2013).Google Scholar
4. Leversha, G., Crossing the bridge, UKMT (2006).Google Scholar