Constructions
143. The Equilateral Blueprint · three sides, three equal arcs
The triangle remains equilateral as you drag A and B.
To construct an equilateral triangle on segment AB: from A and B, draw arcs of radius AB that meet at C. Triangle ABC has all three sides equal to AB by construction — and therefore all three angles equal to 60°.
What this lesson covers
Try to break it
Drag A and B closer or further apart. The two equal-radius arcs always meet at C with AB = BC = CA. Try to make one side different from the others — impossible. Equal arc radii force an equilateral triangle every time.
How you build it
Construct an equilateral triangle.
- Mark point A — one end of the base segment.
- Mark point B — the other end of the base segment.
- Draw segment AB — the base of the equilateral triangle.
- Place compass at A. Set radius equal to AB. Draw an arc above AB.
- Keep the same radius. Place compass at B and draw an arc. Where the two arcs cross is point C.
- Mark point C exactly where the two arcs cross.
- Join A to C — the second side.
- Join B to C — the third side. Triangle ABC is complete!
The proof, step by step
Prove that the constructed triangle is equilateral.
- AB is given as the base of the construction.
- The arc centered at A has radius AB, so the distance AC equals AB.
- The arc centered at B has radius BA, so the distance BC equals AB.
- Therefore, AB = BC = CA. By definition, ΔABC is equilateral.
Worked example
In an equilateral triangle ABC, if AB = 8 cm, what is the length of AC?
By definition, all sides of an equilateral triangle are equal in length. Since AB = 8 cm, AC must also be exactly 8 cm.
- 4 cm
- 6 cm
- 8 cm — correct
- 10 cm