Triangle Centers worksheet
Task #1: Centroid, Circumcenter, Orthocenter Applet
In the GeoGebra applet above, the points D, E, and F represent the centroid, circumcenter, and orthocenter of the triangle ABC. (Not necessarily in that order.) Determine which point is which.
These three centers (Centroid, Circumcenter, Orthocenter) share a special property. In the applet above, move the points A, B, and C to see what changes, and what stays the same over lots of different triangles. Make a conjecture about the points D, E, and F.
Task #2: Location for a new shopping center
What is the geometry name for the location you chose for the shopping center? Why did you make that choice?
Task #3: Location for a Water Treatment Center
Task #3 (a)
Task #3 (b)
The point that minimizes total distance to the vertices of a triangle is called the triangle's Fermat point. Based on your observations, make a conjecture about the Fermat point of a triangle.
Task #4: Find the Fermat point
- Select the "Regular Polygon" tool, then select the points A, B (in that order). When prompted, enter "3" vertices. You have made an equilateral triangle on side AB. (Let's call it ABD.)
- Repeat the last steps to make an equilateral triangle on side BC. (Let's call it BCE.)
- Repeat again to make a third equilateral triangle on side CA. (Let's call it CAF.)
- Construct lines from the third (non-ABC) vertex of each equilateral triangle to the opposite vertex on AB. For example, the first line should pass through points D and C.
- Use the intersect tool to construct the intersection of these lines. This is the Fermat point!