plane

A plane is a two dimensional space that extends indefinitely

point in the plane

A point is an exact localtion in spaces that has no length, width or thickness.

line

A line is a one dimensional object that is infinitely long with no width, depth or curvature

area

An area is a region of a plane

ray

A ray is a line with a single endpoint that extends infinitely in one direction

line segment

Given two points the line segment is part of the line that connects \( A \) and \( B \) so that the line has \( A \) and \( B \) as endpoints. We denote this line by \( \overline{A B} \)

circle

A circle of radius \( r \) around a point \( A \) is the collection of points that are all distance \( r \) away from \( A \)

circle from two points

Given two points \( A , B \) on the plane, then we define the circle from \( A \) to \( B \) as the circle centered at \( A \) with radius \( \text{dist} \left ( A , B \right ) \) and notate it as \( A \odot B \)

compass

A compass is a tool which allows us to contstruct any arc or circle on the plane

straight edge

A straight-edge is a tool which allows us to draw lines, rays or line segments

equilateral triangle construction

We can construct an equilateral triangle using a compass and straight edge

Let \( A , B \) be points on the plane, then we can construct the circle from \( A \) to \( B \), \( A \odot B \) and \( B \odot A \), from this we obtain two intersection points and pick one of them as \( C \), and draw a third circle \( C \odot B \).

We know that \( C \in A \odot B \) so \( \text{dist} \left ( A , B \right ) = \text{dist} \left ( A , C \right ) \), additionally \( C \in B \odot A \) so \( \text{dist} \left ( B , C \right ) = \text{dist} \left ( B , A \right ) \), thus we have \( \text{dist} \left ( A , B \right ) = \text{dist} \left ( A , C \right ) = \text{dist} \left ( B , C \right ) \) and so by drawing line segments between \( A , B , C \) we obtain a triangle all with equal side lengths

And thus the triangle \( \overline{A B} , \overline{B C} , \overline{C A} \) is equilateral.