Challenge 07: Construct an Equilateral Triangle
The construction of an equilateral triangle illustrates some of the most important ideas in dynamic geometry. With this challenge, you will play with that construction. Before working on this challenge with your team, watch a brief YouTube clip that shows clearly how to construct an equilateral triangle: http://www.youtube.com/watch?v=ORIaWNQSM_E
Remember, a circle is all the points (“circumference”) that are a certain distance (“radius”) from one point (“center”). Therefore, any line segment from the center point of a circle to its circumference is a radius of the circle and is necessarily the same length as every other radius of that circle. Even if you drag the circle and change its size and the length of its radius, every radius of that circle will again be the same length as every other radius.
This simple but beautiful example shows the most important features of dynamic geometry. Using just a few points, segments and circles (strategically related), it constructs a triangle whose sides are always equal no matter how the points, segments or circles are dragged. Using this construction, you will know that the triangle must be equilateral (without you having to measure the sides or the angles).
Everyone on the team should construct an equilateral triangle. Play with (drag) the one that is there first to see how it works. Take turns controlling the GeoGebra tools. In GeoGebra, you construct a circle with center on point A and passing through point B by selecting the circle tool, then clicking on point A and then clicking on point B. Then if you drag either point, the size and location of the circle may change, but it always is centered on point A and always passes through point B because those points define the circle.
Euclid began his book on geometry with the construction of an equilateral triangle 2,300 years ago.