金碧辉煌的圣殿 （3. Johnson’s theorem）

The Glorious Temple of Geometry (3. Johnson’s Theorem) 

挂在我家墙上的几何图案中，有一个是在正方形的镜框里面。这个几何图形包含四个圆，（我认为）极其优美。它显示的是Johnson’s theorem。

Among the geometric patterns hanging on the walls of my home, there is one within a square frame. This geometric figure, containing four circles, is exceptionally beautiful, displaying Johnson's Theorem.

《几何原本》写于公元前4世纪，此后的几何规律和定理又被一代代数学家和爱好者不断发现和证明。一些有相当难度、很隐蔽的规律，比如下图所示的“欧拉线定理”，也被欧拉这样的天才数学家在260年前发现并证明了—

The Elements was written in the 4th century BC, and since then, geometric laws and theorems have been continually discovered and proven by generations of mathematicians and enthusiasts. Some challenging and obscure rules, like the "Euler Line Theorem" shown in the figure below, were discovered and proven by geniuses like Euler 260 years ago --

然而这座“金矿”似乎总有开采不完的金子。上一篇我们聊了Morley’s trisector theorem，它是在19世纪的最后一年，即1899年被揭示的。那么，到了20世纪，还能不能有那样简明而优美的发现呢？没问题！

However, this "gold mine" seems to always yield inexhaustible treasures. As discussed in my previous article, we talked about Morley’s trisector theorem, revealed in the last year of the 19th century, 1899. So, can the 20th century bring forth discoveries as concise and elegant? Absolutely!

Johnson’s Theorem 发表于1916年，已经是20世纪了。它是这样表述的：“如果三个半径同为r的圆经过一公共点H，那么经过另外三个交点的圆的半径也是r 【Given three circles of equal radii r that all pass through a common point H, then the circle through their other three intersections has the same radius r.】换句话说，以另外三个交点为顶角的三角形，其外接圆与原先那三个圆是全等的。简单不简单？

Johnson’s Theorem was published in 1916, firmly placing it in the 20th century. It is stated as follows: "Given three circles of equal radii r that all pass through a common point H, then the circle through their other three intersections has the same radius r." In other words, the circumcircle of the triangle formed by the other three intersection points is congruent to the original three circles. Simple, isn't it?

发表这个定理的，是美国数学家Roger Johnson。他1913年获得哈佛大学数学博士学位，不久以后进入纽约的Hunter College数学系。他后来长期担任系主任直到退休。Roger Johnson不是大数学家，他一生仅发表十几篇论文（包括上图中所示的仅一页的关于该定理的论文 A circle theorem。）但他的一部探讨欧式几何复杂问题的专著Modern Geometry - An Elementary Treatise on the Geometry of the Triangle and the Circle (Houghton Mifflin, 1929) 却在几十年里颇欢迎。现在这本书的全文都可以在网上阅读。

The mathematician who published this theorem was the American Roger Johnson. He earned his Ph.D. in mathematics from Harvard University in 1913 and soon after joined the mathematics department at Hunter College in New York. He later served as the department chair until his retirement. Johnson was not a prominent mathematician, and only published just over a dozen papers in his lifetime (including the one-page paper on this theorem shown in the figure). Nevertheless, his book Modern Geometry - An Elementary Treatise on the Geometry of the Triangle and the Circle (Houghton Mifflin, 1929) was well-received for decades and is now available for online reading.

这篇小论文最后一段的讨论，其口气很有意思：我这发现啊，“appear to be new”，可是它如此简单，明确，两千多年来难道所有的人都忽视了？我有点儿心虚啊。读者们，你们如果发现我不是原创，请报告…… 这种论文的写法今天肯定是没有的。

The concluding paragraph of this short paper has an interesting tone: My discovery, "appears to be new," but it is so simple and clear. Could everyone have overlooked it for more than two thousand years? I feel a bit uneasy. Readers, if you find that I am not the originator, please report... Such a writing style is obsolete in academic papers today.

上面两个图看起来有些深奥，其实只是前面那个图稍加扩展。图A显示了第五个圆，它是以三个圆的交点H为圆心，作一个圆，经过另外三个圆的圆心，显然这个圆的半径也是r，所以这个图中五个圆的半径都相同。图B我不细解释了。有一点几何童子功的网友马上就能看出来，那个红色大圆的半径为2r。那三个半径为r的圆无论怎样移动，只要有共点H，它们始终都是与大圆内切的。

The two figures above may seem profound, but they are just a slight extension of the first figure. Figure A shows the fifth circle, which is centered at the intersection point H of three circles, passing through the centers of the other three circles. Clearly, the radius of this circle is also r, so the radii of all five circles in this figure are the same. I won't explain Figure B in detail. Geometry enthusiasts can quickly see that the radius of the large red circle is 2r. The three circles with a radius of r, no matter how they move, as long as they have a common point H, they are always tangent to the large circle.

上面这个图，进一步表现这个几何结构的特点：无论是把六个相关的点连成两个中心对称的全等三角形，还是连成如立方体的透视图，深层次的规律是“万变不离其宗”的。以橙红色代表的Johnson环的半径为r是不变的。

The above figure further illustrates the characteristics of this geometric structure: whether connecting the six related points into two centrally symmetric congruent triangles or connecting them into a perspective view of a cube, the deep-seated rule is "varied but not deviating from its essence." The radius of the Johnson circle, represented in orange-red, remains unchanged.

说到这里，故事并没有完...... Johnson的论文发表后又过了很多年，人们才发现，这个规律其实早在之前八年，即1908年，已经被一个叫Gheorghe Titeica的罗马尼亚人发现并证明了。我本人在知道这件事以后，一开始以为这位Titeica先生是罗马尼亚的某个偏远地区的小镇做题家，没有话语权，让美国佬Johnson占了便宜。

At this point, it seems like it should be the end. However, many years after Johnson published his paper, it was discovered that this rule had already been found and proven by a Romanian named Gheorghe Titeica eight years earlier, in 1908. Initially, I thought Mr. Titeica was a problem solver in a remote town in Romania, with no say, letting the American Johnson take advantage.

我完全错了。Titeica是罗马尼亚著名数学家。他在法国获得博士学位（法国的数学在18-19世纪是世界上最牛的），他是罗马尼亚微分几何（differential geometry）的奠基人，他在罗马尼亚国家科学院长期担任高职，他是罗马尼亚数学学会的主席。作为知名学者，他还是国际数学大会（ICM）几何分支的主席。他发表过几百篇论文，学生中也出了著名数学家，他儿子是知名量子物理学家，而且他本人还是小提琴高手，他牛得很。

I was completely wrong. Titeica was a renowned Romanian mathematician. He earned his Ph.D. in France (French mathematics was the world's best in the 18th and 19th centuries), and he laid the foundation for differential geometry in Romania. He held high positions in the Romanian Academy of Sciences for a long time and served as the president of the Romanian Mathematical Society. As a well-known scholar, he also served as the president of the geometry branch of the International Congress of Mathematicians (ICM). He published hundreds of papers, produced famous mathematicians among his students, and his son was a well-known quantum physicist. Moreover, he was a skilled violinist. He was truly remarkable.

有一天他拿着一枚5-lei 硬币，画了3个共点的圆，突然意识到还有一个全等的圆隐含在里面，他画了出来。至于证明，对他来说实在是小菜一碟。这个平面几何的小问题，简单又直观，他根本不认为是什么大的发现。恰巧这时候罗马尼亚的《数学学报》（Gazeta Matematica）有一个开放性的数学竞赛，让参赛者自己提出命题并证明。当时在布加勒斯特大学当教授的Titeica，就把自己刚刚发现的“the five-lei coin problem”加上证明，用罗马尼亚语撰写后递交上去了。那是1908年。

One day, holding a five-lei coin, he drew three circles with a common point and suddenly realized that another congruent circles were implicit in them, so he drew it. As for the proof, it was a piece of cake for him. This small problem in plane geometry was simple and straightforward, and he didn't consider it a significant discovery. Coincidentally, at that time, there was an open mathematical competition in the Romanian Gazeta Matematica, allowing participants to propose and prove their propositions. Titeica, who was a professor at the University of Bucharest at the time, submitted this "the five-lei coin problem" with proof, written in Romanian.  That was in year 1908.

Titeica教授在数学方面的贡献，根本不以这个“小儿科”发现说事儿。1961年罗马尼亚为他专门发行了一枚邮票，是表彰他在微分几何方面的成就和他的leadership。Titeica也不争什么名分，倒是数学爱好者们常有不平。Roger Johnson虽然是独立发现了相同的规律，但Gheorghe Titeica毕竟比他早8年，也是（用非英语）记录在案的。若称该定理为Titeica-Johnson’s Theorem 应该更公平。我同意这一点。

Professor Titeica's contribution to mathematics does not hinge on this small discovery. In 1961, Romania issued a postage stamp specifically to honor his achievements in differential geometry and his leadership. Titeica did not seek fame on finding and proving the circle problem, but mathematics enthusiasts often express dissatisfaction. Although Roger Johnson independently discovered the same rule, Gheorghe Titeica was eight years ahead, and it was clearly recorded (in a non-English language). Calling the theorem Titeica-Johnson’s Theorem would be fairer. I agree with this.

到了1999年（Titeica已经去世整整60年了），第40届“国际数学奥林匹克”（IMO）在罗马尼亚举行。那神秘的第四个圆再次静悄悄地浮出水面，嵌入大会的Logo（见上）。我认为这个极好的设计 — 第一，第四个圆正好用到“40”里；第二，这是罗马尼亚学者首先发现的；第三，这是属于基础数学范畴，与IMO正相配；第四，定理简明、图形优美、Logo设计相当美观！

In 1999 (60 years after Titeica's death), the 40th International Mathematical Olympiad (IMO) was held in Romania. The mysterious fourth circle surfaced quietly again, embedded in the logo of the event (see above). I think this design is excellent—first, the forth circle fits perfectly into "40"; second, it initially discovered by a Romanian scholar; third, it is on basic mathematics, fitting well with the IMO; fourth, the theorem is concise, the graphic is beautiful, and the logo design is elegant!

【edited from ChatGPT translation】