There is this property of a cubic equation with 3 distinct real solutions where you can draw an equilateral triangle above the graph (center lined up with the inflection point), and it can always be rotated and scaled such that the corners line up with the roots, and the leftmost/rightmost points on an inscribed circle line up with the extrema.
I saw this in a video by Mathologer about the cubic formula and I was wondering if there was a way to solve this type of cubic using the only the triangle and POI/extrema.
I have looked up methods using trigonometry, but they just use algebraic manipulation with the formula for cos(3θ), no direct geometric relation.
I'm not really sure where to start. I was thinking maybe finding the radius of the inscribed circle and relating that to the length of each side, but I wouldn't know which direction to go to reach the corners. Maybe there is some other approach that I am missing.