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Inverted Triangle and Equilateral Triangle are the only triangles which have a finite number of vertices.
My question is as follows:
Inverted Triangle and Equilateral Triangle are the only triangles which have a finite number of vertices. Prove it!
Let $a,b,c$ be the vertices of a finite triangle $A$. Then the rotation $A\to B$ of $A$ where $B$ is the triangle with vertices $a,a+b,a+b+c$ is called symmetry w.r.t. $A$.
Triangles with an even number of vertices are all symmetric w.r.t. their center. So the only triangles with an even number of vertices are rectangles and equilateral triangles.
The idea to reduce to two cases:
Show that the symmetry $A\to B$ can’t have both $a,a+b$ on the outside. This is similar to the following proof.
Show that symmetry w.r.t. $A$ can’t have both $a,a+b,a+b+c$ on the outside. This is similar to the following proof.
For first part: In any triangle with finitely many vertices the sum of their angles is an integer multiple of $\pi$.
The sum of angles of any triangle is $180^\circ$ minus the sum of the angles at the base (which is $180^\circ$ minus the angle at the apex which is $0$). Since triangles with two points on the base are also equilateral and they have an angle of $\pi/2$, it follows that the sum of angles is half of the perimeter of the triangle. Since the perimeter of an equilateral triangle is $2 \sqrt 3$ it follows that the sum of angles of any triangle with finitely many vertices is an integer multiple of $\pi$.
Since any triangle with only finitely many vertices has a finite perimeter it follows from the above that the sum of angles of such a triangle
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