Steiner-Lemus Theorem

|AE|=|BD|,\,\alpha =\beta ,\,\gamma =\delta

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

| AE | = | BD |, \, \ alpha = \ beta, \, \ gamma = \ delta

The Steiner – Lemus theorem is a triangle geometry theorem. It is known as an example with a seemingly simple statement that does not have a simple classical proof, although there is a simple analytical proof.

Content

1 Formulation
2 History of evidence
3 Variations and generalizations
4 Literature
5 notes

Wording

If 2 bisectors are equal in a triangle, then this triangle is isosceles.

Evidence History

The proof was given in the works of German geometers Jacob Steiner and Daniel Lemus .

In 1963, American Mathematical Monthly magazine announced a competition for the best proof of a theorem. A lot of evidence was sent, among which were found interesting previously unknown. One of the best ^[1] , according to the editors, uses the method of the opposite and the circle passing through 4 points as an additional construction.

A proof based on the following criterion for the equality of triangles is widespread in Soviet literature: if an angle, the bisector of this angle and the side opposite to this angle of one triangle are equal to the corresponding elements of another triangle, then such triangles are equal.

The analytical proof follows from the formula for the length of the bisector

l_{c}={\frac {\sqrt {4abp(p-c)}}{a+b}}.

{\ displaystyle l_ {c} = {\ frac {\ sqrt {4abp (pc)}} {a + b}}.}

Variations and generalizations

A similar theorem for bisectors of external angles (segments of bisectors of external angles drawn before the continuation of the sides) is false. One of the counterexamples is the Bottem triangle - with angles of 12 °, 132 ° and 36 °. In it, the segments of the bisectors that are external to the first two angles drawn before the intersection with the extensions of the sides are equal to the side connecting their vertices.

Literature

Optional course in mathematics. 7-9 / Comp. I. L. Nikolskaya. - M .: Education , 1991 .-- S. 335-338. - 383 p. - ISBN 5-09-001287-3 .
Ponarin I.P. Elementary geometry. In 2 volumes - M .: ICSTMO , 2004 .-- S. 31. - ISBN 5-94057-170-0 .
Weisstein, Eric W. Steiner – Lehmus theorem on Wolfram MathWorld .
Some Proofs of the Steiner-Lemus Theorem

Notes

↑ Coxeter G. S. M. , Greitzer S. P. New encounters with geometry . - M .: Nauka , 1978. - T. 14. - ( Library of the mathematical circle ).

[1] Coxeter G. S. M. , Greitzer S. P. New encounters with geometry . - M .: Nauka , 1978. - T. 14. - ( Library of the mathematical circle ).