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More variations on Nagel and Gergonne analogues of the Steiner-Lehmus theorem

Published online by Cambridge University Press:  23 August 2024

Sadi Abu-Saymeh  and
Mowaffaq Hajja
Sadi Abu-Saymeh
Affiliation:
2271 Barrowcliffe Drive, Concord, NC 28027, USA e-mail: ssaymeh@yahoo.com
Mowaffaq Hajja
Affiliation:
P. O. Box 388, 21510 Al-Husun, Jordan e-mail: mowhajja@yahoo.com, mowhajja1234@gmail.com
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Extract

The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal then the corresponding sides have equal lengths. That is to say if P is the incentre of ΔABC and if BP and CP meet the sides AC and AB at B′ and C′, respectively, then

An elegant proof of this theorem appeared in [1] and is reproduced in [2].

Type
Articles
Information
The Mathematical Gazette , Volume 108 , Issue 572 , July 2024 , pp. 292 - 302
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Copyright
© The Authors, 2024 Published by Cambridge University Press on behalf of The Mathematical Association

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References

Gilbert, G. and MacDonnel, D., The Steiner-Lehmus theorem, Amer. Math. Monthly 70 (October 1963) pp. 79–80.CrossRefGoogle Scholar
Abu-Saymeh, S., Hajja, M., More variations on the Steiner-Lehmus theme, Math. Gaz. 103 (March 2019) pp. 1–11.CrossRefGoogle Scholar
Sastry, K. R. S., A Gergonne analogue of the Steiner-Lehmus theorem, Forum Geom. 5 (2005) pp. 191–195.Google Scholar