GeoGebra Lab #3: Triangle Centers
Task #1: Centroid, Circumcenter, Orthocenter Applet
Question 1
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.
Question 2
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
Question 3
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)
![Toolbar Image](/images/ggb/toolbar/mode_angle.png)
Question 4
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
Instructions:
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!