《金碧輝煌的聖殿（4. Never say never）》The Magnificent Temple (4.)

金碧輝煌的聖殿（4. Never say never）The Magnificent Temple (4.)

在歐氏幾何這兩千多年的發展史上，共圓（cocircular）問題一直是一個十分有趣的話題，這方麵的大量的定理和命題被發現，極大地豐富了人們對圖形的認知。稍有幾何知識的人都知道，三點如果不在一條直線上，則總是共圓的。四點若要共圓則需要一些條件，比如四點構成的四邊形的對角的角度之和為180度，這是四點共圓的充分必要條件。

In the more than two thousand years of the development of Euclidean geometry, the problem of cocircularity has always been an intriguing topic. Numerous theorems and propositions in this area have been discovered, significantly enriching people's understanding of geometric shapes. Anyone with a basic knowledge of geometry knows that if three points are not on the same line, they are always cocircular. For four points to be cocircular, certain conditions are required, such as the sum of the angles formed by the diagonals of the quadrilateral being 180 degrees, which is a sufficient and necessary condition for four points to be cocircular.

若是更多的點共圓，有時情況就比較“邪乎”了。例如非常有趣且著名的“九點圓定理”（nine-point circle theorem）是法國數學家Olry Terquem 在幾代人結果的基礎上，於200年前最終揭示的。九點圓定理的表述也非常簡潔：“任意三角形三高線的垂足、各邊中點、各頂點與垂心連線的中點，這九個點共圓。”（The midpoint of each side of the triangle, the foot of each altitude and the midpoint of the line segment from each vertex to the orthocenter all lie on the same circle. ）

When it comes to more points being cocircular, things can get a bit "weird." For instance, the fascinating and famous "Nine-Point Circle Theorem" was revealed by the French mathematician Olry Terquem about 200 years ago, building on the results of several generations. The theorem is succinctly stated as follows: "The midpoint of each side of the triangle, the foot of each altitude, and the midpoint of the line segment from each vertex to the orthocenter all lie on the same circle."

很神奇是不是？其實遠遠不止於此。比如，這個“九點圓”的半徑是同一個三角形外接圓半徑的一半；這個圓分別與同一個三角形的三個旁切圓外切。而且，這個圓的圓心一定在歐拉線上。上一篇我們提到了歐拉線。現在有人誤以為歐拉也發現了九點圓。其實不是的，他生前沒有看出來。

Isn't it amazing? In fact, it goes far beyond this. For instance, the radius of this "Nine-Point Circle" is half the radius of the circumcircle of the triangle. This circle is also tangent to the three excircles of the same triangle. Moreover, the center of this circle must lie on the Euler line, which was mentioned in the last article of this series. Contrary to popular belief, Euler did not discover the Nine-Point Circle during his lifetime.

對中國的幾何愛好者來說，名氣最大的不是上麵那個九點圓，而是1838年由法國數學家Auguste Miquel 發現並證明的的“五點共圓定理”（Miquel's Pentagram Theorem ）。結合下圖，這個定理的表述也非常簡明：任意一個五角星形（不要求正五角星），分別對其五個三角形做外接圓，則外側的五個交點共圓。(There are five triangles externally to the pentagon. Construct the five circumcircles. Each pair of adjacent circles intersect at a vertex of the pentagon and a second point. The five second points are concyclic.)

For geometry enthusiasts in China, the most famous theorem is not the Nine-Point Circle mentioned above, but the "Five-Point Cocircularity Theorem" discovered and proven by the French mathematician Auguste Miquel in 1838. With reference to the diagram below, the theorem is succinctly stated: "For any pentagram (not necessarily regular), construct the circumcircles for each of its five triangles; then, the outer five intersection points are cocircular." 

這個定理之所以今年在中國大陸有名，是因為一年前去世的前國家主席江澤民。他是工科出身，也是一位數學愛好者。據他說，求證“五點共圓定理”是他當年考大學的一道題。（水平了得！）於是他多次拿出來考人。他接見中國IMO隊時提到這道題，還打電話到中科院院士張景中家裏討論這道題，弄得老張一腦門子汗。最有名的一次是2000年12月20日，江主席在澳門出席澳門特別行政區成立一周年慶祝活動，並參觀濠江中學。他即興給同學們出了這道題，並親自到黑板上畫圖。濠江中學師生很少做這麽難的題，聖誕都沒過踏實，全校攻關，終於在12月28號得證，算過了一個踏實的新年。其實早在5年前，哈佛大學數學係教授、菲爾茲獎獲得者丘成桐訪問中國時，江主席就從兜裏麵摸出這道題讓他證明。老邱一看就暈菜了，心想您這是讓將軍挖戰壕啊。他回到賓館閉門苦思，終於能在第二天離京前能夠給中辦一條信息：請轉告江主席，他出的題我證出來了，Oh yeah！！！

This theorem gained prominence in mainland China this year because of the passing of former Chinese President Jiang Zemin. With a background in engineering, he was also a mathematics enthusiast. According to him, proving the "Five-Point Cocircularity Theorem" was a question in his university entrance exam (such a high level!). He frequently presented this problem to others. He mentioned this problem when he met Chinese IMO team, and he even discussed it with academician Zhang Jingzhong of the Chinese Academy of Sciences, causing some stress for Zhang. One notable instance occurred on December 20, 2000, when President Jiang, during his visit to Macau for the celebration of its first anniversary as a Special Administrative Region, spontaneously posed this problem to students at Ho Fai High School, and personally drew the diagram on the blackboard. The students at the High School were not accustomed to tackling such challenging problems, and their Christmas was far from peaceful as the entire school collaborated to solve it. Eventually, on December 28, they succeeded, marking a satisfying end to the year. In fact, five years earlier, when Harvard University mathematics professor and Fields Medalist Shing-Tung Yau visited China, President Jiang pulled out this problem from his pocket and asked Yau to prove it. Yau, feeling a bit overwhelmed, returned to his hotel room and, after intense contemplation, was able to send a message to the Central Office the next day: Please convey to President Jiang that I have proven the problem he posed. Oh yeah!!!

這些事情，有時都造成了國際影響了。比如濠江中學出題這事兒，韓國媒體就報道了：您瞧人家中國主席多睿智、多有學問。再看看俺們青瓦台的那幾塊料，除了玩政治還會啥？…… 去年江主席去世的時候，很多人，包括我本都反省，當年我們笑話他愛賣弄，抨擊他獨裁，有點不公平啊。他肚子裏有貨，有資格賣弄嘛。而他當政那時候畢竟給我們一點抨擊、說怪話空間啊…… 不怕不識貨，就怕貨比貨！

These occurrences sometimes create international impact. For instance, when the problem from Ho Fai High School was publicized, Korean media reported: Look at how wise and knowledgeable the President of China is compared to our Blue House, those in it know how to play politics. Last year, when President Jiang passed away, many people, including myself, reflected on the fact that we had mocked him for showing off and criticized him for being authoritarian. It seems a bit unfair now. He had substance, and he had the qualifications to show off. During his presidency, he at least gave us some room for criticism and bizarre remarks… It's not about not recognizing quality; it's about comparing quality!

扯遠了。我們回來聊幾何。我們前兩次說的Morley’s trisector theorem 和 John’s theorem 分別是在19和20世紀才被發現的。雖然歐氏幾何就像一座特殊的金礦，總能有耀眼的金塊被發現，但到了21世紀，在歐氏幾何被人類學習、研究了2400年後，人們還能發現簡單的幾何規律嗎？

You bet it！

Let's get back to geometry. The two theorems we discussed in my earlier articles, Morley's Trisector Theorem and John's Theorem, were discovered in the 19th and 20th centuries, respectively. Although Euclidean geometry is like a special gold mine where dazzling nuggets are always discovered. As we entered the 21st century, after 2400 years of humans studying and researching Euclidean geometry, can we still find simple geometric laws?

You bet it!

時間到了2000年。在荷蘭的南部海濱有個叫Zeeland的小地方，一位年輕的高中數學老師叫Floor van Lamoen。本來中學老師照本宣科就行了，但這位van Lamoen先生是個數學迷，愛琢磨事兒，他時常在歐氏幾何和初等數論方麵有些小進展、小發表（不是大數學家那種）。我們都知道任何三角形頂角與對邊的中點連成中線，三條中線交於重心G，把三角形分成麵積相等的6塊。他對6個小三角形分別做外接圓，驚奇地發現6個圓的圓心似乎是共圓的。他將三角形的形狀作了多種變換，從圖形上看，這個規律幾乎肯定是存在的。

Fast forward to the year 2000. In the southern coastal region of the Netherlands, there is a small place called Zeeland, where a young high school math teacher named Floor van Lamoen resides. Usually, high school teachers follow the curriculum, but Mr. van Lamoen is a math enthusiast who likes to ponder. He often makes small advances and publications in Euclidean geometry and elementary number theory (not at the level of renowned mathematicians). We all know that connecting the vertex of any triangle with the midpoint of the opposite side forms a median. The three medians intersect at the centroid G, dividing the triangle into six regions of equal area. He constructed circumcircles for the six small triangles and was amazed to find that the centers of the six circles seemed to be cocircular. He experimented with various transformations of the triangle's shape, and from a graphical perspective, this pattern almost certainly existed.

然而在試圖證明時，van Lamoen卻被卡住了，努力好幾個月居然未能攻克。這個六點共圓問題的確是不太好證的。相比較而言，前麵那個“九點圓定理”倒是比較好證。

However, when attempting to prove it, Mr. van Lamoen got stuck and, despite months of effort, couldn't overcome the challenge. The problem of six points being cocircular was indeed not easy to prove. Comparatively, the earlier "Nine-Point Circle Theorem" was easier.

於是van Lamoen考慮再三，決定把它作為一個open problem征答。他將問題登在了American Mathematical Monthly 2000年11月號的Problems欄目中。在閱讀這個問題的人中間，有一個人叫 Kin-Yin Li，中文名叫李健賢。他當時是香港科技大學的年輕數學教授，幾年前剛剛從加州伯克利拿了數學博士學位，回港加盟香港科技大學數學係。這位李博士是個解難題的人。而且 此前幾年，IMO有一年在香港舉行。借著這個東風，包括李健賢在內的一些促進香港競賽數學的人士創辦了一個競賽數學的News Letter，叫Mathematical Excalibur。當時李健賢是它的編輯和重要撰稿人，後來他成了香港競賽數學的教父級人物。

So, after serious consideration, van Lamoen decided to pose it as an open problem. He published the problem in the "Problems" section of the November 2000 issue of the American Mathematical Monthly. Among those reading the problem was a person named Kin-Yin Li, also as 李健賢in Chinese. At the time, he was a young mathematics professor at the Hong Kong University of Science and Technology, having recently earned a Ph.D. in mathematics from the University of California, Berkeley. Dr. Li is someone skilled at solving problems. Moreover, in the previous years, an International Mathematical Olympiad (IMO) had been held in Hong Kong. Taking advantage of this, Dr. Li and other enthusiasts promoting mathematical competitions in Hong Kong established a newsletter called "Mathematical Excalibur." At that time, Dr. Li served as its editor and a significant contributor. Later, he became a key figure in the world of mathematical competitions in Hong Kong.

李健賢馬上就看到了van Lamoen的問題。這個問題難不住他，他很快得到證明。的確，證明不是很簡單，他用了幾乎一頁紙的篇幅。他的證明登在了2001年春天的一期Mathematical Excalibur上，正式成為了定理。現在這個圓依然叫“Van Lamoen Circle”，李博士並沒有表麵上的名分，他也不需要。有趣的是（見下圖），他在解這道題之前，先提到了江澤民主席在澳門濠江中學出的那道題。該題與van Lamoen問題的提出時間相近，李博士也注意到了，於是先評論一下。

Dr. Li immediately noticed van Lamoen's problem. It didn't stump him, and he quickly provided a proof. Indeed, the proof was not very straightforward, taking up almost a full page. His proof was published in the spring issue of Mathematical Excalibur in 2001, officially becoming a theorem. The circle associated with this discovery is still called the "Van Lamoen Circle," and Dr. Li did not seek surface recognition for it, nor did he need to. Interestingly (as seen in the image below), before tackling this problem, Dr. Li mentioned the problem posed by President Jiang Zemin at in Macau. The timing of this problem's proposal was close to that of van Lamoen's, and Dr. Li took note of it, providing some comments beforehand.

那麽，今後人們還會繼續有這樣的發現嗎？應該說，歐氏幾何發展到今天，新發現的機會越來越小了，但可能性依然是存在的。也許將來某一天，一名愛好者或者業餘數學家，在不經意之間，能再次揭示出某一條簡明而優美的歐氏幾何規律……

Never say never.

So, will people continue to make such discoveries in the future? It must be said that as Euclidean geometry has been studied and researched by humans for 2400 years, the chances of new discoveries are diminishing. However, the possibility still exists. Perhaps one day, an enthusiast or amateur mathematician will, inadvertently, reveal another concise and elegant Euclidean geometric law...

Never say never.

[Edited from ChatGPT translation]