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.

Tick your answer here

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.

  • Hint: Trying holding a ruler up to the screen.

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)

Move the point D to a position that minimizes Total Distance. Drag the point D around until you get "Total distance" as small as possible. Hint: The minimum total distance is very close to 17 units.

Task #3 (b)

To explore the position that made this minimization possible, measure the angles ADB, BDC, and CDA. Toolbar ImageTo find the angles, click the "Angle" tool, then select, IN ORDER, the points A, D, B. Then, select B, D, C. Then, select C, D, A.

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:

  1. Toolbar Image 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.)
  2. Toolbar ImageRepeat the last steps to make an equilateral triangle on side BC. (Let's call it BCE.)
  3. Toolbar ImageRepeat again to make a third equilateral triangle on side CA. (Let's call it CAF.)
  4. Toolbar ImageConstruct 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.
  5. Toolbar ImageUse the intersect tool to construct the intersection of these lines. This is the Fermat point!