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.