Explorations with GSP
Consider any triangle ABC. Find a construction for a point P such that the sum of the distances from P to each of the three vertices is a minimum.
The point we are looking for is called the Fermat Point.
In order to find this point we first construct a triangle ABC and choose any arbitrary point D. It does not matter whether this point D is inside or outside of triangle ABC. No matter where point D is placed, the resulting point P will always be in the same location.
Next, we create segments from point D to each vertex of the triangle.
We then rotate triangle ABD 60 degrees counter-clockwise and triangle ACD 60 degrees clockwise.
We then connect point BÕ to point C, and point BÕÕ to point A. The intersection of these points is where point P is.
To make sure point P is always in the same location, letÕs try again placing point D inside the triangle.
Once again, we create segments from point D to each vertex of the triangle.
We then rotate triangle ABD 60 degrees counter-clockwise and triangle ACD 60 degrees clockwise. And connect BÕ to C and BÕÕ to A.
We see that point P stays in the same place. Therefore, this is the only construction of point P that results in the minimum distance from P to each of the vertices.