geometry

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

\overset{―}{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

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 $C \in A ⊙ B$ so $dist (A, B) = dist (A, C)$ , additionally $C \in B ⊙ A$ 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 $\overset{―}{A B}, \overset{―}{B C}, \overset{―}{C A}$ is equilateral.

🏗️ ΘρϵηΠατπ🚧 (under construction)

🏗️ $Θ ρ ϵ η Π α τ π$ 🚧 (under construction)