I'm trying to write an article series on ''Impossible-looking mathematical theorems". I may not be familiar to many of this kind of mathematical theorems. So I need a suggestion of which to include. The theorems should be externally beautiful. You can also suggest me some solved problems. As an example, Basel problem can be one of them because it states that the infinite sum containing natural numbers includes π, which looks quite impossible and wonderful. Please help me.
2026-04-03 22:55:38.1775256938
Mathematical theorems which look impossible externally
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Around 1900, we not only explored alternatives to Euclidean geometry, but also had an explosion of discoveries in it. Euclidean geometry contains countless theorems that boil down to, "this extremely specific-sounding property, which you'd think would apply to few if any triangles (or occasionally some other shape), applies to them all". You need to learn a lot of them just to understand why the nine-point centre even exists. In fact, there are tens of thousands of triangle centres which are non-obviously well-defined for all triangles. You'll learn many beautiful results, some pertaining to circles or other polygons, from books such as this or this.