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The Fermat point of a triangle, also called the Torricelli point, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible. Now, we describe two ways to construct a Fermat point.
Step 1. Construct an equilateral triangle on each of two arbitrarily chosen sides of the given triangle.
Step 2. Draw a line from each new vertex to the opposite vertex of the original triangle.
Step 3. The two lines intersect at the Fermat point.
Exercise. Try to prove that the above construction is valid. When you are ready, read below.
There is a classic problem where the solution is a Fermat point. This problem is called the Steiner tree problem. The Fermat point is also applicable in many other geometric problems including some beautiful olympiad problems.