The Math Forum || Annie's Sketchpad Activites

Exploring the Euler Segment Using Script Tools

In this activity you'll create scripts that construct the centroid, circumcenter, and orthocenter of a triangle and use them to investigate the segment that connects those points.

The Centroid

  1. Using the segment tool, construct a triangle. Drag each vertex of your triangle to make sure that all the segments are connected properly.
  2. Select two segments (using the shift key) and select "Point at Midpoint" from the Construct menu.
  3. Use the segment tool to draw the two medians, as in the picture below on the left.
  4. Using the selection arrow, click on the intersection of the two segments, as in the picture on the left.
  5. Using the text tool (the hand) click on this new point. A label should appear. Still using the text tool, double-click on the label itself (NOT the point) and you will get box that will let you relabel the point. Rename it centroid and click okay.
  6. Select the two medians and the two midpoints and select "Hide Objects" from the Display menu. Your picture should look like the one on the right. Move the vertices of your triangle to make sure that the centroid behaves as it should.

Making a Script

Make sure your selection arrow is highlighted and choose "Select All" from the Edit menu. Under the Work Menu choose "Make Script". You will get a script box on the right of your screen. Choose "Save AsÉ" from the File menu and save your script on your floppy. Name the file "centroid".

Note: You can save your script tool wherever you want, as long as you put them all together and you know where to find them!

Choosing a Script Tool Folder

  1. Choose "Preferences" under the Display menu.
  2. Click on the "More" button.
  3. Where it says "Script Tool Folder" in the middle of the screen select "Set..."
  4. Navigate to find your floppy. If you're on a Macintosh, click the "Select (nameofdisk)" button at the bottom of the dialog box. If you're using a Windows machine, make sure that it says a: and nothing else.
  5. Click "Continue" and then on "Okay".

You should now have another tool at the bottom of your tool palette that looks like an arrow. This is your script tool. Click on it now and select "centroid" from the menu that appears. You are now holding a centroid tool in your hand! Click on three places on the screen (without dragging). You get a triangle and a centroid!

Try using the segment tool to construct a triangle, then use your centroid tool to construct the centroid of that triangle.

Constructing the Circumcenter and the Orthocenter

Now we want to construct the other two special points that we will be investigating.

  1. Start a new sketch and construct a triangle using the segment tool.
  2. Construct two altitudes of the triangle and select the point of intersection.
  3. Use the text tool to label this point "orthocenter".
  4. Hide the altitudes, select everything in your sketch, and choose "Make Script" from the Work menu. Save this script as "orthocenter" in the Workshop folder.
  5. Do the same thing for the circumcenter, constructing the perpendicular bisectors of two of the sides, finding their intersection, and labeling it "circumcenter". Make the script and save it in the Workshop folder.
  6. Now when you hold down the script tool in the tool palette, you should have three things available to you. Practice using each of them on different triangles, then move the triangles around to see how they interact.

Exploring the Euler Segment

We now have built a "library" of tools that we can use to investigate the Euler Segment. These tools are available to us at any time, and might just represent a small part of the library we will develop throughout the year.
  1. In a new sketch, use the segment tool to construct a triangle.
  2. Use your centroid, orthocenter, and circumcenter tools to construct those special points of this triangle. Drag a couple of vertices of your triangle to see what happens.
  3. Use the segment tool to construct the segment between the circumcenter and the orthocenter. Drag your triangle again.
  4. Measure the distance from the circumcenter to the centroid, and the distance from the orthocenter to the centroid. Notice anything? Move your triangle around some more.
  5. Calculate the ratio of the distance measurements.