Part 1: Construction of the triangle's incenter

• Draw an arbitrary triangle ABC and drag each of the vertices until the triangle appears to be an equilateral triangle.  Students should use either angle measurement or the length of sides to verify the status of their triangle.
  
  

• Visualize the three interior angle bisectors of the triangle.  Obviously any two of the angle bisectors will intersect inside the triangle.  Where will the third bisector lie?  What other conjectures can you predict about the third angle bisector?  Make a sketch of any triangle and its angle bisectors to assess your prediction.  
   
  

• In the sketch window, construct the bisectors of <A and <B and their point of intersection.  Construct the bisector of <C.  How does this resulting construction compare with your previous prediction?

• Now suppose we have a scalene triangle instead of an equilateral triangle.  What would you predict about the angle bisectors?  
  
  

• Drag any vertex or side of your equilateral triangle to form a scalene triangle.  Observe what happens to the intersection of the angle bisectors.  How does this fit with your prediction?  
   
  

• Further manipulate the triangle by dragging any of the triangle’s vertices.  Take note of what happens to the point of intersection.  Discuss and form a written conjecture that describes your findings.

The point of intersection of the three angle bisectors is called the incenter of the triangle.

• Focus your attention on the distances from the incenter to each of the three sides of the triangle.  Make a general statement about their relationship.  
  
  

• Measure the distances from the incenter to each of the three sides along the angle bisector. To do this you will need to find the point of intersection of each angle bisector and the corresponding opposite side.  Manipulate your triangle by dragging any vertex or side.  How are these distances related? 

Part 2: Finding the location of an oil pumping station

• Can you help Lambeth Oil Company determine where the pumping station will reside?  Click and drag the map to a different location in your sketch window, so you may easily see triangle ABC.  Provide a sketch and a written description how you determined the pumping station’s location. You may assume the ground is flat and the drawing is to scale.  (Instructor Note: The map and solution have been saved as pipeline.gsp.)

• To make it easier to distinguish between the angle bisectors and the lines perpendicular to the sides change the color of the perpendicular lines in your sketch window.

• Copy and paste your triangle into a new sketch window.  Manipulate your triangle and observe the perpendicular distances from the incenter to the sides. Make a generalization about the perpendicular distances from the incenter to the sides of any triangle. 
  
  
• Manipulate your triangle so that the perpendicular lines coincide with the angle bisectors.  Classify your the triangle. 
  
  

• Formally prove that the perpendicular distances from the incenter to each of the three sides of any triangle are equal.
  
  

• Inscribe a circle in the triangle that stays inscribed as you manipulate the triangle.  In a new sketch window, define a new custom tool to inscribe a circle in a triangle.  Write a description of your construction process.

Part 3: Finding the location of a school bus garage

• Open the sketch highschl.gsp. In the sketch window, you will see the figure below.

Albemarle, Monticello, and Western Albemarle high schools are planning to build a new bus garage to house all of the county school buses. After much debate on where to build the garage, the school board has decided to build the garage at a point equal distance from each of the three schools. 

• Find the location of the new school bus garage for Albemarle County. Click and drag the map to a different location of your sketch window, so you may easily see the triangle ABC. Provide a sketch and a written description how you determined the location of the garage. You may assume the map is to scale. Write a short report to the school board either in support or against their decision for the location of the garage. Be sure to comment of the feasibility of the location you found. (Instructor Note: The map and solution has been saved schools.gsp.)

The point of intersection of the three side bisectors is called the circumcenter of the triangle.

• Copy and paste your triangle into a new sketch window.  As you manipulate your triangle, does the circumcenter always remain inside the triangle?  Where does the circumcenter of an equilateral triangle lie?  An isosceles triangle?  A right triangle? 
  
  

• Measure the distances from the circumcenter to each of the vertices of the triangle.  Write a conjecture describing your findings.
  
  

• Formally prove that the distances from the circumcenter to each of the three vertices of any triangle are equal.

Extensions:

• It is projected in the year 2015 that Albemarle County will need to build another high school due to population growth.  How can you generalize your solution to find the location of the garage when there are four high schools in Albemarle County?
  
  
• In what kind of quadrilateral can you inscribe a circle?  Try to inscribe a circle in a few different quadrilaterals.  Generalize a statement about in what kinds of quadrilaterals can you inscribe a circle.  What about other polygons?  Can you inscribe a quadrilateral in any polygon?