compass and straightedge construction of regular triangle
One can construct a regular triangle with sides of a given length $s$ using compass and straightedge as follows:

1.
Draw a line segment^{} of length $s$. Label its endpoints $P$ and $Q$.

2.
Draw an arc of the circle with center $P$ and radius $\overline{PQ}$.

3.
Draw an arc of the circle with center $Q$ and radius $\overline{PQ}$ to find a point $R$ where it intersects the arc from the previous step.

4.
Draw the regular triangle $\mathrm{\u25b3}PQR$.
This construction is justified by the following:

•
$\overline{PQ}\cong \overline{PR}$ since they are both radii of the circle from step 2;

•
$\overline{PQ}\cong \overline{QR}$ since they are both radii of the circle from step 3;

•
Thus, $\mathrm{\u25b3}PQR$ is an equilateral triangle^{};

•
In Euclidean geometry^{}, any equilateral triangle is automatically a regular triangle. Therefore, $\mathrm{\u25b3}PQR$ is a regular triangle.
This construction is based off of the one that Euclid provides in The Elements as the first proposition^{} of the first book. Please see http://planetmath.org/?op=getmsg;id=15600this post for more details.
This construction also yields a method for constructing a ${60}^{\circ}$ angle using compass and straightedge.
Note that, with the exception of actually drawing the sides of the triangle^{}, only the compass was used in this construction. Since regular triangles tessellate, repeated use of this construction provides a way to find infinitely many points on a line given two points on a line using just a compass.
If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.
Title  compass and straightedge construction of regular triangle 

Canonical name  CompassAndStraightedgeConstructionOfRegularTriangle 
Date of creation  20130322 17:19:10 
Last modified on  20130322 17:19:10 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  10 
Author  Wkbj79 (1863) 
Entry type  Algorithm^{} 
Classification  msc 51M15 
Classification  msc 5100 
Synonym  compass and straightedge construction of equilateral triangle 
Synonym  compass and straightedge construction of equiangular triangle 
Defines  compass and straightedge construction of ${60}^{\circ}$ angle 