# Equilateral Triangle and Euclid’s First Proposition

The first theorem in Euclid’s Elements is about the construction of an equilateral triangle. To construct equilateral triangle $ABC$, let $AB$ be a line segment.  Construct a circle with center $A$ passing through $B$. Next, construct another circle with center $B$ passing through $A$. Let $C$ be one of the intersections of the two circles as shown in the following figure. Draw segments $AC$ and $BC$ to complete the triangle. The key to the proof that $ABC$ is an equilateral triangle is the fact that congruent circles have congruent radii. The proof is very elementary as shown below.

The Theorem

Given the two circles, one with center $A$ passing through $B$, and another circle with center $B$ passing through $C$, and their interesection $C$, $ABC$is an equilateral triangle.

The Proof

Congruent circles have congruent radii. It follows that AB latex\$, $AC$, and $AD$ are congruent since they are radii of  congruent circles. We know that equilateral triangles have congruent sides, so $ABC$ is equilateral.