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.