geometrymath-historyopen-problemsoft-question
Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history.
What I would like to know is: What is the oldest open problem in geometry?
Also (soft questions): Why is it so hard? Which existing tools may be helpful to handle it? If twenty great geometers of today gathered to work together in the problem, would they (probably) be able to solve it?
P.S. The problem can be of any area of geometry (discrete, differential, etc…)
I suggest you not try to stick to some specific order. If something interests you one day then read it. Forcing yourself to stick to a schedule will only cause you to be unmotivated and as a result not get as much out of it. Also, if your going to self study, try and still talk with people about what your doing or make videos in which your try to explain what you've learned. If you really want a challenge test your progress by trying to do problems here on MSE. I hope this helps and good luck. Also, search around for an ebook which shows you some elementary proof techniques because mathematics at this level requires a lot more time and understanding.
$\newcommand{\Reals}{\mathbf{R}}$Theorem: Let $S^{2} \subset \Reals^{3}$ be the unit sphere centered at the origin, and $n = (0, 0, 1)$ the north pole. The stereographic projection mapping $\Pi:S^{2} \setminus\{n\} \to \Reals^{2}$ is conformal.
Proof: Fix a point $p \neq n$ arbitrarily, and let $v$ and $w$ be arbitrary tangent vectors at $p$. The plane containing $n$ and $p$ and parallel to $v$ cuts the sphere in a circle $V$ tangent to $v$. Similarly, the plane containing $n$ and $p$ and parallel to $w$ cuts the sphere in a circle $W$ tangent to $w$.
The circles $V$ and $W$ form a digon with vertices $p$ and $n$. By symmetry, the angle $\theta$ they make at $p$ is equal to the angle they make at $n$. Let $v'$ and $w'$ be the tangent vectors at $n$ obtained from $v$ and $w$ by reflection across the plane of symmetry that exchanges $p$ and $n$.
Because the tangent plane to the sphere at the north pole is parallel to the $(x, y)$-plane, the image $\Pi_{*}v$ of $v$ under $\Pi$ is parallel to the vector obtained by translating $v'$ along the ray from $n$ through $p$ to the $(x, y)$-plane. Similarly, $\Pi_{*}w$ is parallel to the vector obtained by translating $w'$.
It follows at once that the angle between $\Pi_{*}v$ and $\Pi_{*}w$ is equal to the angle between $v'$ and $w'$, which is equal to the angle between $v$ and $w$.
Conformality of stereographic projection allows the holomorphic structure on the Riemann sphere to be visualized in three-dimensional Euclidean geometry, providing a basic link between complex analysis and differential geometry.
In coordinates, stereographic projection and its inverse are given by $$ \Pi(x, y, z) = \frac{(x, y)}{1 - z},\qquad \Pi^{-1}(u, v) = \frac{(2u, 2v, u^{2} + v^{2} - 1)}{u^{2} + v^{2} + 1}. $$ It's possible to calculate the induced Riemannian metric on the plane: $$ g = \frac{4(du^{2} + dv^{2})}{(u^{2} + v^{2} + 1)^{2}}. $$ Conformality is encoded in $g$ being a scalar function times the Euclidean metric. While this analytic argument has its own elegance, the geometric argument above is essentially obvious.
Best Answer
Problem:
For acute triangles, the question has been answered affirmatively by Fagnano in 1775: one can simply take the length $3$ orbit joining the basepoints of the heights of the triangle.
For (generic) obtuse triangles, the answer is not known in spite of very considerable efforts of many mathematicians. Apparently, A. Katok has offered a $10.000$\$ prize for a solution of this problem.