straightedge-and-compass construction

Absolutely! Here's how to construct an equilateral triangle using just a compass and a straight ruler:

Steps:

Draw a Line Segment: Start by using your straightedge to draw a line segment of any desired length. This segment will represent the side length of your equilateral triangle.

Set the Compass: With your compass, set the point (sharp end) on one endpoint of the line segment you just drew. Then, adjust the opening (distance between the point and the pencil lead) to match the length of the line segment. Essentially, you're using the line segment itself to set the compass width.

Draw Circles: Without changing the compass width, place the point on each endpoint of the line segment. Make two arcs - one above the line segment from each endpoint. These arcs will intersect above the line segment.

Connect the Intersections: Now, take your straightedge and connect the point where the two arcs intersect with each endpoint of the original line segment.

Explanation:

By setting the compass width to the length of the line segment and using that width to draw arcs from both endpoints, you ensure both arcs have the same radius. The point where the arcs intersect is the same distance away from both endpoints because the compass width remained constant. Therefore, the two lines you draw from the intersection point to the endpoints will each be the same length as the original line segment, forming an equilateral triangle with all three sides congruent.

Bonus Tip:

Since all the sides of an equilateral triangle are congruent, the three interior angles are also congruent, each measuring 60 degrees. So, by constructing an equilateral triangle, you've also created a tool to measure 60-degree angles!