Let's start the construction by drawing a segment using a straight edge.
This can be of any length.
The sides of equilateral triangles are congruent, so the distance of the third vertex from A and B is the same as AB.
We can find the position using a compass.
First draw an arc with center A that goes through B.
Draw a second arc, this time with center B that goes through A.
The intersection point of these two arcs is the third vertex of the equilateral triangle.
Verifying the Construction
We can verify the construction by measuring the sides of the triangle.
We can see that the lengths of the three sides are the same, so this is indeed an equilateral triangle.
You will probably have different measurements, but you should also get the same measure for all three sides.
Justifying the Construction
To justify the construction we need to show that AB=BC=CA.
Claim
Justification
CA=AB
Both B and C are on the arc we constructed first with center A. The length of AB and CA is the radius of the same arc, so they are equal.
BC=AB
Both A and C are on the second arc we constructed with center B. The length of AB and BC is the radius of the same arc, so they are equal.
Since both BC and CA is the same as AB, this is indeed an equilateral triangle.
