△ ABC is ANY acute triangle (i.e. all three angles are less than 90o). Equilateral triangles are drawn on each side of △ ABC. They are shown in the diagram as red △ BCY, green △ ACZ, and blue △ ABX.
Straight lines are then drawn joining A to Y, B to Z and C to X. Those lines intersect at F - the FERMAT point - named after Pierre de Fermat (1601 - 1665) who is probably better known for the so-called "Last Theorem" proved in the early 1990s by Professor Andrew Wiles.
The Fermat Point is sometimes called the Fermat - Torricelli point - (after Evangelista Torricelli 1608 - 1647).
The Fermat Point is the point in the triangle from which the sum of the distances to the vertices is a minimum. That is, where AF + BF + CF is smaller than the sum of the distances from any other point in the triangle to the vertices.
It can
readily be proved that AY = BZ = CX. All that is required is to show that pairs of triangles are congruent. Consider △ ABZ and △ ACX.AC = AZ because △ ACZ is equilateral
AB = AX because △ ABX is equilateral
∠ CAX = ∠ BAZ ... each angle is 60o + ∠ BAC
Hence △ ABZ and △ ACX are congruent and BZ = CX.
In a similar way it can be proved that BZ = AY.
Rotation
If △ ABZ is rotated by 60o about point A, then Z would move to point C and B would move to point X.
Angles at the Fermat Point
Each angle at the Fermat Point (e.g. ∠ AFB) is equal to 120o
Proof of concurrency
There are various proofs that AY, BZ, CX are concurrent at the Fermat Point (F). These are not included in this post.
Circumcircles
The circumcircles of the three equilateral triangles are drawn. They all pass through the Fermat Point.
Material
ScientificLib.com - Fermat Point
Dynamicmathematicslearning - Fermat-Torricelli Point Generalization
Youtube - Video - Brilliant.org/ThinkTwice - The Fermat Point - a very clear presentation
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