Is there a point $O$ inside a triangle $\triangle ABC$ (any triangle) such that the angle $\angle{AOB} = \angle{BOC} = \angle{AOC}$?
What do we call this point?
Geometry – Does a Triangle Always Have a Point Where Each Side Subtends Equal 120° Angles?
euclidean-geometrygeometrytrianglestrigonometry
Best Answer
This is not the case for every triangle, $1^\circ-1^\circ-178^\circ$ triangle, for example, is one of the counterexamples to this claim. However, if all angles are less then $120^\circ$, then the claim is true.
To construct such a point; Take any side $[AB]$, find two intersections of perpendicular bisector and circle with radius $\dfrac{|AB|}{2\sqrt 3}$ centered at middle point of $[AB]$. Call this points $A'$ and $B'$. Draw two circles contains points $A'AB$ and $B'AB$. All the $120^\circ$ angles that see $[AB]$ are on these circles. If you apply these procedure to other sides and take intersection points of these circles, you can see combinations of intersection points such that three circles intersect, gave you two points. One of these points are always outside of the triangle and you can see other point could be outside or inside of the triangle.
Update: Apparently; these two intersection points are named Fermat points; point on always outside is called second Fermat point, and the other is called first Fermat point. Also, above circles which have these points on are called Vesica piscis.
Here is a picture of these Fermat points and circles: