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 B to point C, and point B to point A. The intersection of these points is where point P is.


 

To make sure point P is always in the same location, lets 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 B to C and B 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.


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