A polygon having three sides of equal length and three vertices of equal angle (always 60 degrees).

The regular tetrahedron, the regular octahedron and the regular icosahedron all have faces which are equilateral triangles.

This idea lets you construct a 60 degree angle at a point A with just compass and straight edge. It's pretty simple.

  1. Make a line starting at A, and extend it to B.
  2. Open your compass to some distance smaller than AB.
  3. Put the compass point at A.
  4. Draw a small arc which intersects AB. Call the point at which it crosses C.
  5. Put the compass point at C.
  6. Draw a circle with the compass.
  7. Put the compass point at A.
  8. Draw an arc which crosses the circle. Call the point where it crosses D.
  9. Draw the straight line AD. Bingo!

This works because you have just defined three points (A, C and D) of an equilateral triangle!

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