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For a particular case of a figure called simplex, a triangle is surprisingly complicated (in my opinion).

As an illustration, see the list of triangle topics on Wikipedia, and the Triangle page.

The high-school (or even university) curriculum hardly touches on a half of theorems and facts related to triangles (in my experience), unless the student's speciality is related to geometry.

Is it possible that there are still some unexplored properties of Euclidean triangles? Are there known examples of still unproved conjectures in this area?

Edit

The topics in other fields (such as number theory), which originate/are closely connected with the geometric properties of triangles are also included in this question

Yuriy S
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    There probably are undiscovered facts, but the problem lies in the first comment of this post: http://math.stackexchange.com/questions/661553/why-are-there-so-few-euclidean-geometry-problems-that-remain-unsolved . Namely that any first order statement in Euclidean geometry can be mechanically proven or disproven by a computer. So interesting conjectures that remain unsolved necessarily must deal with subtler aspects such as topology, for the actual geometry itself is a theory that can now be totally automated – Sidharth Ghoshal Jul 08 '16 at 23:31
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    A large class of geometric problems is in principle algorithmically solvable by coordinatizing and using the decision procedure for real-closed fields. However, things change a great deal if the problem is about existence of certain configurations in which certain specified dimensions are rational. – André Nicolas Jul 08 '16 at 23:33
  • @frogeyedpeas, what about complicated topics such as coloring? It's not geometry, but if it's about Euclidean triangles, this relates to my question – Yuriy S Jul 08 '16 at 23:41
  • @frogeyedpeas, also, it's surely possible that there is some property of triangles, that no one has ever considered, so even if it's easy to prove, it would still be something new, right? – Yuriy S Jul 08 '16 at 23:46
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    coloring, is, again a different beast. As for "there is some property of triangles that no one has considered", yes there are infinitely many! And they would be new, but finding something tasteful or compelling to the lay-mathematician (if there's such a thing?), is the hard part – Sidharth Ghoshal Jul 08 '16 at 23:50

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As André Nicolas commented, there are open problems involving rational distances. For example, is there a triangle whose sides, altitudes and medians are all rational?

Community
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Robert Israel
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The Kobon triangle (https://en.wikipedia.org/wiki/Kobon_triangle_problem) problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura. The problem asks for the largest number $N(k)$ of nonoverlapping triangles whose sides lie on an arrangement of $k$ lines.

There's also circle-packing in an equilateral triangle (https://en.wikipedia.org/wiki/Circle_packing_in_an_equilateral_triangle), and circle-packing in an isosceles right triangle (https://en.wikipedia.org/wiki/Circle_packing_in_an_isosceles_right_triangle) and it's probably worth having a look at Klee and Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory.

Under problem D19 in Guy, Unsolved Problems In Number Theory, 3rd edition, page 286, the unsolved problem is stated: "which integers occur as the ratios base/height in integer-edged triangles?

Gerry Myerson
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  • Thank you! The last problem statement is extremely unclear. Is this the full quote? Does the problem ask to list all such integers? Or to find out some unifying property of them? Or to prove there is only finite/infinite number of them? – Yuriy S Jul 09 '16 at 01:45
  • Well, I left out the close quote at the end, but other than that, that's the full statement of the question. It's followed by some discussion. I'd surmise the problem is to characterize said integers, but maybe that's asking too much. – Gerry Myerson Jul 09 '16 at 01:54