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斯坦纳定理:如果三角形的两条角平分线相等,则为等腰三角形。这个命题是由Lehmus于1840年提出来的,瑞士的几何家斯坦纳(1796—1863)首先给出了一个证明。本文介绍一种能为初中文化水平的读者所接受的证法,及一个更为普遍的定理。已知:在△ABC中 (图一),AD、BE是角平分线,且AD=BC 求证:AC=BC。证明:假设AC≠BC。 (1)若AC∠CBA,∠CAD>∠CBE。在∠DAC内作∠DAN=∠CBE,AN必落在∠DAC的内部,设AN分别交BE、BC
Steiner’s theorem: If the bisectors of the two angles of the triangle are equal, they are isosceles triangles. This proposition was put forward by Lehmus in 1840, and the Swiss geometrician Steiner (1796-1863) first gave a proof. This article describes a proof method that can be accepted by readers of junior high school culture level, and a more general theorem. It is known that in △ABC (Figure 1), AD, BE are the angle bisectors, and AD=BC Proof: AC=BC. Proof: Assume AC ≠ BC. (1) If AC ∠ CBA, ∠ CAD> ∠ CBE. In the ∠DAC, ∠DAN=∠CBE, AN must fall within the ∠DAC, and set AN to BE and BC respectively.