Bonjour Geometry! (Part 1)

Find a ruler (tool #1), but disregard its markings. You will not measure with it, just draw lines.  Find a compass (tool #2) to draw circles. You will not use it to transfer distances or for measuring. Now you have all the tools that Euclid had to build an equilateral triangle.

Build your own equilateral triangle. Are you certain it is equilateral? You can prove it. Do not measure the lengths of its sides, but use what you know about the characteristics of circles to show the three sides are congruent, and therefore the triangle is equilateral.

 

Answer

Draw a segment of any desired length. Label its end points B and C, then place the tip of your compass at B and draw a circle with center at B and radius BC.

Then place the tip of your compass at C and draw a circle with center at C and radius BC.

Euclid Prop 1 vr2 step1

Let A be the top intersection point of the two circles. Use your ruler to connect B to A and C to A.

Is triangle ABC equilateral?

Euclid Prop 1 vr2

 

Segments BC and BA are both radii of the same circle (with center at B), therefore are congruent:

BC ≡ BA

Segments CB and CA are also radii of the same circle (with center at C), and so they are congruent:

CB ≡ CA

In Euclid’s words “things which equal the same thing also equal one another,” and we have:

BC ≡ BA ≡ CA, and triangle ABC is equilateral.

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