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#GeoGebra Tutorial 2A – Constructing an Equilateral Triangle with Compass & Straightedge

#2 
#3 
Problem: How do you draw an equilateral triangle using compass and straightedge?
This construction is almost exactly the same as #2 except that here we will exactly mimic compass and straight edge construction using the compass tool. The idea is to use the intersections of two circles and the two centers to form the triangle as shown below.
Click & drag A, B (or a=AB or the triangle itself). Notice that the triangle is always equilateral!
Figure 1  An equilateral triangle formed by radii of two circles
Construct online!

1.) Click the Segment between two points tool, and click two distinct places on the drawing pad to construct segment AB. 

2.) If the labels of the points are not displayed, click the Move tool, right click each point and click Show label from the context menu. (The context menu is the popup menu that appears when you right click an object.) 

3.) To construct a circle with radius AB and center A, click the inverted triangle on the Circle toolbar, select the Compass tool, click point A, then click point B and then click point A again. After step 3, your drawing should look like the one shown in Figure 2 . We have included the Algebra View so that you can see the objects created. However since the position of your points is probably different, the values of your objects will be too. 

Figure 2  Circle with center A and passing through B.


4.) To construct another circle now with radius AB and center B, with the Compass tool still active, click point A,then point B and then center point B.


5.) Next, we have to intersect the circles. To intersect the two circles, click the inverted triangle on the Point toolbar, select Intersect Two Objects tool, then click on the top intersection point of both circles. After step 5, drawing should look like the one shown in Figure 3. 

Figure 3  Circles with radius AB and intersection C and D.


6.) To make the construction easier to see, we will change the properties of the 2 circles. Rightclick one of the circles, click Object Properties from the dropdown menu.
The Object Properties window will open.


7.) In the Object Properties window. At left, click on category Circle. At right, select the Style tab. Click on the inverted arrow by Line Style and click on ......... Click on Close.
Figure 4  Object Properties for circles


8.) We make the other 2 sides of the triangle. Click on the Segment tool and then on B and C.


9.) With the Segment tool still active, click on C and then A. After step 9, drawing should look like the one shown in Figure 5.


Figure 5  Triangle formed from radii of two circles


10.) Use the Move button, move the vertices of the triangle. What do you observe? Notice that you can move vertex A and B but you can never move vertex C. This is because vertex C is a dependent object. Recall that vertex C is the intersection of two circles and thus depends on the length of segment AB. 

11.) You have probably observed that it seems that ABC is an equilateral triangle. In fact, it is. To verify, we can display the lengths of the sides of the triangle.
Rightclick one of the sides of the triangle, click Object Properties from the dropdown menu.
The Object Properties window will open.


12.) In the Object Properties window. At left, click on category Segment. At right, select the Basic tab. If Show label is not selected, select it.
Click on the inverted arrow by Show label and click on Value. Click on Close.
Figure 6  Object Properties for segments


Figure 7  Triangles with side lengths


13.) Prove that the construction above always results to an equilateral triangle. 
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