ΘρϵηΠατπ

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 AB
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 dist(A,B) and notate it as A⊙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⊙B and B⊙A, from this we obtain two intersection points and pick one of them as C, and draw a third circle C⊙B.

We know that C∈A⊙B so dist(A,B)=dist(A,C), additionally C∈B⊙A so dist(B,C)=dist(B,A), thus we have dist(A,B)=dist(A,C)=dist(B,C) and so by drawing line segments between A,B,C we obtain a triangle all with equal side lengths

And thus the triangle AB,BC,CA is equilateral.