Constructing An Equilateral Triangle

HOW TO CONSTRUCT THE DIAGRAM FROM THE ELEMENTS, BOOK I, PROPOSITION 1

In this post, we will show you how to construct an equilateral triangle, when you are given nothing but a line segment to start. In the process, we will construct the same diagram found in Euclid’s The Elements, Book I, Proposition 1. This is also the same diagram used in our earlier post “Proving A Triangle Is Equilateral”.

Step 1: We use a straightedge to create a line segment. We name one endpoint A, and the other endpoint B. Not surprisingly, we name this line segment AB.

Step 2: We use our compass to construct a circle around A. We push the pin of the compass into point A, and adjust the compass until the pencil lies on point B. In doing this, we have set the compass to construct a circle with a radius equal to the length of AB. While keeping the pin of the compass pressed into point A, we gently spin the compass around. As the pencil drags across the paper, it draws out our first circle! We have now constructed a circle centered at point A, with radius AB.

Step 3: We then pick up the compass, and push the pin into point B. Line up the pencil of the compass with point A; it should already be the correct radius. We will now use the compass to draw another circle, this one centered at B and with BA as a radius. When we are finished, we will have two identical, overlapping circles who centers are connected by the line segment AB.

Step 4: See that point above line segment AB where the two circles intersect? Let’s call that point C.

Step 5: Now, we’ll draw a line segment from point A to point C. Since both AB and AC are radii of circle A, they must be the same length. We’ll put a small tick mark on each one to remind us of this congruency.

Step 6: Finally, we will draw our last line segment, from point B to point C. Since both BA and BC are radii of circle B, they must be the same length. We’ll put a small double tick mark on each one to remind us of this congruency.

Step 7: At this point, we have now constructed a triangle. As we have proven earlier, this triangle ABC must be equilateral. I’ve outlined the triangle in bright blue, marked each side with triple tick marks, and lightly shaded the inside of the triangle.

Congratulations! You have now constructed a perfectly equilateral triangle, without ever having to use a ruler to measure length, or a protractor to measure angles.