What relationships are created in the construction of the equilateral triangle? Explore some of the relationships that are created among line segments in this more complicated figure. What line segments do you think are equal length – without having to measure them? What angles do you think are equal? Try dragging different points; do these equalities and relationships stay dynamically? Can you see how the construction of the figure made these segments or angles equal?
When you drag point F, what happens to triangle ABF or triangle AEF? In some positions, it can look like a different kind of figure, but it always has certain relationships.
What kinds of angles can you find? Are there right angles? Are there lines perpendicular to other lines? Are they always that way? Do they have to be? Can you explain why they are?
Can you prove why triangle ABC is always equilateral (see hint on proof)?