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.

equilateral 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, ABCis 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.

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