Part 1: Construction of the triangle's centroid

• Three brothers, George, Glenn and Gary recently inherited their father’s triangular plot of fenced farmland in southern Virginia, shown in the figure below. As directed in their father’s will, they are to divide area of land equally between them. The brothers agree to share the cost to divide the land into three separate fenced plots using the least amount of fencing. Each brother wishes to have both of their brothers as neighbors, so the three plots must be adjacent. Discuss with your partner how the brothers should go about dividing the land equally to minimize the amount of fencing needed?

• Open the sketch land.gsp. In the sketch window, you will see the figure above.  
   

• We’ll begin our exploration with a similar, simpler problem. What if there were only two brothers, how would you divide the triangular plot of land in half? (Hint: Area = ½ base· height.)
  
  

• In your sketch window, click and drag the map to a different location of your sketch window, so you may easily see triangle ABC.  Construct the midpoint of the base of the triangle. Construct a segment connecting the midpoint to the vertex opposite of the triangle’s base. This segment is called a median of a triangle. How does this construction compare with your previous idea to divide the triangle’s area in half?

• Confirm, by measuring, that the segment divided the triangle’s area in half. Manipulate your triangle. Do the areas of the triangles remain equal as you manipulate the original triangle in your sketch window?

• Write and formally prove a conjecture depicting the previous construction.  
  
  

• How can you use the construction of the triangle’s median to help George, Glenn and Gary divide their father’s plot of land into three equal parts?
  
  

• Visualize the triangle’s three medians and predict where they will intersect. Construct and find the point of intersection of the triangle’s three medians. Drag any vertex or side of your triangle to form another triangle. Observe what happens to the medians’ point of intersection. Write a conjecture describing your findings.  

The point of intersection of the three medians is called the centroid of the triangle.

 Part 2: Dividing a triangle's area into three equal parts

• Investigate the relationship between the areas of the interior triangles formed by the three medians of the triangle. State and prove a conjecture about this relationship.

• How can we use your conjecture to help George, Glenn and Gary divide their father’s triangular plot of farmland?
  
  
• Investigate the relationship between the perimeters of the interior triangles formed by the three medians in the triangle. State a conjecture about their relationship.

• Divide the triangular area of land equally between George, Glenn and Gary. Write a convincing argument that your proposed division of the land uses the least amount of fencing. (Instructor’s Note: The map and solution have been saved as land2.gsp.)  

Part 3: Investigating the medians of a triangle

• Copy and paste your triangle into a new sketch window.  Focus your attention on one of the medians. Make a statement about the relationship between the distance from the vertex to the centroid and the distance from the centroid to the side of the triangle.
  
  
• Measure the distances from the vertex to the centroid and from the centroid to the side of the triangle. How are these distances related?

• Manipulate your triangle, and make a conjecture about this relationship. What is the relationship between the distance from the vertex to the centroid and the distance from the centroid to the side of the triangle?
  
  

• Formally prove the centroid in a triangle divides each median into two parts, the ratio whose lengths are 2:1.

Part 4: Construction of the triangle's orthocenter

We have constructed the incenter, circumcenter and centroid of a triangle by constructing the angle bisectors, side bisectors and medians of the triangle, respectively.

• Continuing this investigation, in a new sketch window construct the altitudes of each of the sides of an arbitrary triangle. Find the point of intersection of the three altitudes. Drag any vertex or side of your triangle. Observe what happens to the altitudes point of intersection. Write a conjecture describing your findings.
  
  

• The point of intersection of the three altitudes is called the orthocenter of the triangle. Does the orthocenter always remain inside the triangle? Where does the orthocenter of an equilateral triangle lie? An isosceles triangle? A right triangle?

Extensions:

• In the presentn directory, open the file called totaltri.gsp. This dynamic sketch allows you to investigate the incenter, circumcenter, centroid, and orthocenter of many different types of triangles. To activate a button, simply double-click on it. To cease the spinning button, click anywhere inside your sketch window. Explore the different centers of a specific triangle.
  
  
• Investigate the Euler line, and define the Euler line in your own words. Investigate the Euler line for right, isosceles and equilateral triangles. Conjecture the placement of the Euler line of all right triangles, all isosceles triangles, and all equilateral triangles.