In Kiselev's Planimetry book, he asks the construction problem: In a given triangle, find a point from which its sides are seen at the same angle.
I'm somewhat sure that this involves using the construction of an arc that encloses a given angle through a segment (On a given segment AB, to construct a disk segment enclosing a given angle), which is proven in the text and which I've used for a majority of the other construction problems. I also know that there is no "given angle" here, but that it would have to be 120 degrees because, for the angles around the necessary point were all equal, it'd be 360/3 = 120, but I can't construct a 120 degree angle.
That's the information I have an I am stumped as to what to do, or how to even put it together. My intuition tells me that I won't be using the 120 degree fact at all, and instead will have to somehow construct three circles that all intersect at the necessary point, but I'm not certain what properties the circles should have.
What's your problem with constructing a $120^\circ$ angle?
Anyway, you can find the Fermat point as follows: On each side, erect (to the exterior) an equilateral triangle. From the third vertices of these triangles, draw the circles through the other two vertices of the same equilateral triangle. These circles intersect in the desired point (provided no angle of the originally given triangle exceeds $120^\circ$).