Are there some undiscovered/unproved theorems about Euclidean triangles?

Question

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?

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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

Answer 1

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?

Answer 2

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?