Tour 4: Investigating Triangle Centers

In this tour, you'll continue honing your skills at geometric construction by constructing a triangle's centroid and other classical triangle centers. Then you'll harness Sketchpad's power in two important new ways: first, by turning your constructions into custom tools, then by making a multi-page document. 
 

What You Will Learn

• How to change Sketchpad's Preferences

• How to construct a triangle's centroid and at least one other classical triangle center

• How to turn a construction into a custom tool and how to use that tool to quickly reproduce the construction

• How to show the Script View of a custom tool and how to step through a script

• How to create a multi-page Sketchpad document

Constructing a Centroid

1. Set up Sketchpad so that points are automatically labeled when constructed.
   To do this, choose Preferences from the Edit menu, go to the Text panel, and check For All New Points. Before clicking OK, you may want to explore what other options are available in Preferences.

2. Use the Segment tool to construct a triangle.

3. Construct the midpoints of the three sides.
   A shortcut: With the Segment tool active, choose Select All Segments from the Edit menu. Then choose Midpoints from the Construct menu; all three midpoints are constructed at once.

4. Use the Segment tool to construct just two of the triangle's three medians. (A median connects a vertex with the midpoint on the opposite side.)

5. Using the Point tool or the Selection Arrow tool, click at the intersection of the two medians. This constructs the point of intersection.

6. Label the intersection point "Centroid." Before clicking OK, check Use Label In Custom Tools. (You'll see why soon.)

7. Construct the third median.

You're ready to make a conjecture: The three medians in a triangle intersect in a single point. But is that true in all triangles?

8. Use the Drag Test to test your conjecture.
   You can also select the entire figure and choose Animate Objects from the Display menu.

Turning It into a Custom Tool

It doesn't take too long to construct a centroid, but if you needed to construct a lot of them for different triangles, it would be nice to do so quickly. Custom tools allow you to do just that, as you'll see in the next section.

9. Select the entire figure. (Either choose Select All from the Edit menu or use a Selection Rectangle.)

10. Choose Create New Tool from the Custom Tools menu (the bottom tool in the Toolbox). Name the tool Centroid and click OK.

11. Click on the Custom Tools icon to choose your new tool. Now click three times in the sketch plane. You should get a construction just like the original one.

12. Use the tool a few more times, clicking sometimes in blank space and other times on existing points or segments. Choose Undo Centroid and then Redo Centroid from the Edit menu and notice that these undo and redo the entire construction at once. Notice also that while most of the points are given new labels, the centroid is always labeled Centroid. This is because of what you did in step 6.

13. Choose the Selection Arrow tool and click in blank space to deselect all objects. Then choose Show Script View from the Custom Tools menu. You'll see a window called Centroid Script. This is a readable description of the construction performed by the custom tool Centroid. Take a moment to look through the script and see how it relates to the construction.

14. Move the Script View window off to the side (by dragging it by its title bar) but don't close it. Select three points in the sketch. All three Givens should be highlighted in the Script View window and you should see two new buttons at the bottom of the window.

15. Click on the Next Step button as many times as it takes to get through the entire script. "Stepping" through a script is a great way to understand how an unfamiliar tool works or to check student work.

Creating a Multi-Page Document

When you visit a web site, it's rare to see all the content on one page -- it's usually distributed over many linked pages. Similarly, in Sketchpad, you can create multi-page documents with several pages of related content. Students can use multi-page documents to create portfolios of their Sketchpad work.

16. Clean up your sketch a bit, perhaps by Undoing back to the point at which there was just one triangle with its medians and centroid.

17. Choose Document Options from the File menu. In the dialog box that appears, choose Blank Page from the Add Page popup menu, then click OK. You’ll now be on page 2 of a two-page Sketchpad document.

18. Click on the tab marked "1" at the bottom of the window to go back to the previous page, then return to the new, blank page.

19. Choose one of the three remaining classical triangle centers listed below. Construct it on page 2 of your document using the Toolbox tools and Construct menu commands. Then turn your construction into a new custom tool, just as you did with the centroid.

Circumcenter: The point of concurrency (intersection) of the perpendicular bisectors of the three sides of a triangle.
Incenter: The point of concurrency of the three angle bisectors in a triangle.
Orthocenter: The point of concurrency of the three altitudes in a triangle.

Further Challenges

• The centroid divides each median into two smaller subsegments. Use measurements to explore the length ratio of any pair of subsegments.

• The three medians of a triangle divide it into six smaller triangles, each defined by one vertex of the original triangle, one midpoint of an original side, and the centroid. Construct these six interiors, then measure their areas. What do you notice? Why is this true?

•  If you constructed a circumcenter in step 19, use it to construct the circumcircle of its triangle -- the unique circle that perfectly circumscribes the triangle.

•  If you constructed an incenter in step 19, use it to construct the incircle of its triangle -- the unique circle that perfectly inscribes the triangle.

• If you constructed an orthocenter tool in step 19, use it to explore the following question: Where is the orthocenter of any triangle defined by two of the vertices and the orthocenter of another triangle? Why?

• Continue adding pages and tools to your document until there are four pages and four tools -- one for every classical triangle center.

• Once you've completed the previous challenge, apply each of the four tools to a single triangle. Three of the four centers will lie along the same segment, called the Euler segment. Which three centers are these? Use measurements to explore the distance relationships between these three centers.