straightedge-and-compass construction

Building an equilateral triangle using only a compass and a straightedge is a classic exercise in geometric construction. Here’s a step-by-step guide on how to do it:

1. Draw the Base Line:

   • Start by drawing a straight line segment using the straightedge. This will be one side of the equilateral triangle. Let’s call the endpoints of this segment (A) and (B).

2. Draw Arcs to Find the Third Point:

   • Place the compass' point at one endpoint of the line segment, say (A), and adjust its width to the length of the line segment (AB).
   • Without changing the compass width, draw an arc above the line segment. This arc will be part of a circle centered at (A) with radius (AB).
   • Now, move the compass to the other endpoint, (B), and draw another arc that intersects the first arc. The intersection point of these two arcs is equidistant from (A) and (B), ensuring that (AB), (AC), and (BC) can all be the same length.

3. Draw the Remaining Sides:

   • Use the straightedge to draw a line segment from (A) to the intersection point of the arcs. This is the second side of the triangle.
   • Draw another line segment from (B) to the intersection point. This completes the equilateral triangle.

4. Finalize the Equilateral Triangle:

   • You now have an equilateral triangle (ABC) where all sides (AB), (AC), and (BC) are of equal length, and all angles are 60 degrees.

This construction is based on the principle that the set of all points a fixed distance (the length of (AB)) from a given point (either (A) or (B)) forms a circle. The intersection of the circles centered at (A) and (B) is equidistant from (A) and (B), ensuring that all sides of the triangle are of equal length, which is the defining characteristic of an equilateral triangle.