Some geometric theorems are well proven, but inverse theorems are difficult to prove. To reconcile the conditions and conclusions of a proposition is to reverse the proposition. If both the proposition and the inverse proposition are proved, it becomes a theorem.
The equality of the two bottom-angle bisector lines of the isosceles triangle is well proven, but conversely, the proof of the inverse theorem is difficult. More than 2,000 years ago, Euclid gave proof of this theorem in The Primitives of Geometry. The inverse theorem, the Steiner–Remios theorem, is it true that a triangle with two equal angular bisection lines is an isosceles triangle?
Euclid knew that the inverse theorem was valid, but suffering from no way to start, failed to prove this simple geometric fact in the "Primitive Geometry", and handed over the white paper.
After 2000, the problem turned around. The 19th-century mathematician Lehmus specifically pointed out that this is a seemingly simple but difficult problem to start. The question was raised in a strong interest of the famous Swiss geometrist Steiner, who gave the first proof. Later, for more than 100 years, hundreds of methods of evidence were given to the Steiner–Remios theorem.
The origin of the problem
Screenshot of the new concept geometry on 63 pages
As shown in the figure, △ABC is known to be an isosceles triangle, and it is easy to prove that the two bottom-angle bisector lines BP and CQ are equal. Just prove that △APB and △AQC are equal.
However, knowing that the bottom-angle bisector BP and CQ are equal, it is difficult to prove that AB=AC. A simple proof can be obtained using the problem-solving tool provided by the new concept geometry, the common angle inequality (generalized common angle theorem). For details, please see the link below: https://m.toutiao.com/is/FKsKUrN/?= Solving Geometry Problems with Area Relations: A (Last) Children's Day Gift for Seventh Graders - Today's Headlines
The last few pictures in the link are transcripts of Mr. Zhang Jingzhong's proof.
The first proof
Everything is difficult at first, and the first one proves to be difficult, which also reflects Steiner's talent.
In 1840, the mathematician Lehmus discovered the proposition that a triangle equal to two inner angle bisector lines is an isosceles triangle. It was difficult to prove it in pure geometry, so he wrote to Sturm, who in turn provided the question to some mathematicians, the First to answer the Swiss geometer Steiner. Hence the problem known as the "Steiner–Remios theorem".
Steiner's first proof, through geometric rotational transformations, subtly deduces conclusions.
Certificate 1: Figure 1. If BD and CE are the angular bisector of △ABC, if ∠ B ≠ ∠C, you may want to set ∠B > ∠C, then ∠DBC > ∠ECB, ∠BDC > ∠BEC.
∴ DC > EB ...... （1）
Also known as △ E'BC ≌ △ECB (i.e. rotation transformation), then
∠BDE'= ∠BE'D 、 ∠CDE' > ∠EC'D
∴ CD < CE'=EB ...... （2）
(1) There is a clear contradiction with (2), and the opposite ∠B < ∠C is not true, so there is ∠B = /∠C. □
The introduction of the self-theorem aroused great interest among European mathematicians, and from 1840 to 1855, at least 15 methods of evidence were discovered, and many articles were written on this issue in various journals of 1842, 1844, 1848, and 1854-1864.
Obviously, direct proof is more difficult, so people began to seek simple proof of the theorem. Around 1940, based on the French mathematician Rebaffet's lemma that "the bisector of the large angles in the triangle is smaller", someone used the counter-proof method to give a simpler proof. But the fly in the ointment is the lemma, which is as difficult as the theorem. After entering the middle of this century, people became more interested in the theorem, and in a comprehensive newspaper in the 1960s, it was pointed out that there were more than 60 kinds of evidence for theorems. By the 1980s, theorems spread around the world.
First, it is widely regarded as the simplest method of evidence, which was published in Canada's Conundrum around 1980
The degree of simplicity is not only astonishing: what is important is its method of proof, which can be fully extended to the "Steiner's theorem" in non-European geometry.
Unexpected surprise
This theorem would definitely be included in the Dictionary of Mathematical Titles, so I thought of consulting the data. The result is not to check do not know, check there are surprises, it is really beneficial to open the book!
In the dictionary, this entry is called "Staeiner-Lehmus theorem." The article was co-authored by Mr. Zhang Jingzhong and Tu Rongbao.
Let's take a look at the nice proof that the dictionary provides. The proof below was presented by two British engineers, G. Gilbert and D. Macdonnell, published in the American Mathematical Monthly in 1963.
By consulting the data, not only did you get a beautiful proof, but you also got the formula for calculating the bisector of the inner angle of the triangle. It was an unexpected surprise and got more than you expected.
In addition to this, there is also a calculation formula, please see the following figure:
Science has not yet been popularized, and the media still needs to work. Thanks for reading, goodbye.
I wish you all a safe and healthy Dragon Boat Festival! [Teeth]