Geometry for elementary school/Constructing equilateral triangle

Introduction

In this chapter, we will show you how to draw an equilateral triangle. What does "equilateral" mean? It simply means that all three sides of the triangle are the same length.
Any triangle whose vertices (points) are A, B and C is written like this: $\mathrm{△}ABC$.
And if it's equilateral, it will look like the one in the picture below:

The construction

The construction (method we use to draw it) is based on Book I, proposition 1.

1. Using your ruler, Draw a line whatever length you want the sides of your triangle to be.
   Call one end of the line A and the other end B.
   Now you have a line segment called $\overline{AB}$.
   It should look something like the drawing below.
   
   
   
   
2. Using your compass, Draw the circle $\circ A,\overline{AB}$, whose center is A and radius is $\overline{AB}$.
   
   
   
   
3. Again using your compass Draw the circle $\circ B,\overline{AB}$, whose center is B and radius is $\overline{AB}$.
   
   
   
   
4. Can you see how the circles intersect (cross over each other) at two points?
   The points are shown in red on the picture below.
   
   
   
   
5. Choose one of these points and call it C.
   We chose the upper point, but you can choose the lower point if you like. If you choose the lower point, your triangle will look "upside-down", but it will still be an equilateral triangle.
   
   
   
   
6. Draw a line between A and C and get segment $\overline{AC}$.
   
   
   
   
7. Draw a line between B and C and get segment $\overline{BC}$.
   
   
8. Construction completed.

Claim

The triangle $\mathrm{△}ABC$ is an equilateral triangle.

Proof

1. The points B and C are both on the circumference of the circle $\circ A,\overline{AB}$ and point A is at the center.
   
   
   
   
2. So the line $\overline{AB}$ is the same length as the line $\overline{AC}$.
   Or, more simply, $\overline{AB}=\overline{AC}$.
   
   
   
   
3. We do the same for the other circle:
   The points A and C are both on the circumference of the circle $\circ B,\overline{AB}$ and point B is at the center.
   
   
   
   
4. So we can say that $\overline{AB}=\overline{BC}$.
   
   
   
   
5. We've already shown that $\overline{AB}=\overline{AC}$
   
   
   and $\overline{AB}=\overline{BC}$.
   
   
   
   Since $\overline{AC}$ and $\overline{BC}$ are both equal in length to $\overline{AB}$, they must also be equal in length to each other.
   So we can say $\overline{AC}=\overline{BC}$
   File:Geom eqtr proof10.png
   
   
   
6. Therefore, the lines $\overline{AB}$ and $\overline{AC}$ and $\overline{BC}$ are all equal.
   
   $\overline{AB}=\overline{AC}=\overline{BC}$
   
   
   
   
7. We proved that all sides of $\mathrm{△}ABC$ are equal, so this triangle is equilateral.

Problems with the proof

The construction above is simple and elegant. One can imagine how children, using their legs as compass, accidentally find it.

However, Euclid’s proof was wrong.

In mathematical logic, we assume some postulates. We construct proofs by advancing step by step. A proof should be made only of postulates and claims that can be deduced from the postulates. Some useful claims are given name and called theorems in order to enable to use them in future proofs.

There are some steps in its proof that cannot be deduced from the postulates. For example, according the postulates he used the circles $\circ A,\overline{AB}$ and $\circ B,\overline{AB}$ doesn’t have to intersect.

Though that the proof was wrong, the construction is not necessarily wrong. One can make the construction valid, by extending the set of postulates. Indeed, in the years to come, different sets of postulates were proposed in order to make the proof valid. Using these sets, the construction that works so well on the using a pencil and papers, is also sound logically.

This error of Euclid, the gifted mathematician, should serve as an excellent example of the difficulty in mathematical proof and also the difference between proof and our intuition.