# GeoGebra Lab #3: Triangle Centers

- Author:
- Dr. Abigail Brackins

## 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.

*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

**minimize**the total distance from the treatment center, D, to each of the three towns. Complete the tasks listed below this applet.

## Task #3 (a)

## Task #3 (b)

**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:

- 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!