135. The 60° Lock · How matching arcs lock into a perfect angle

Try to break it

Drag O anywhere on the canvas. The whole construction follows, but the angle at O always reads exactly 60°. The two equal-radius arcs form an equilateral triangle, so the 60° is locked in. Try to break it.

How you build it

Construct a 60 degree angle.

• Place point O — the vertex of the 60° angle you will construct.
• Place point A to the right of O. A marks the direction of the base arm ray OA.
• Ray tool: click O, then click A. This draws ray OA — the base arm of the 60° angle.
• Arc tool: click centre O, then click out to set a radius r. The arc swings across ray OA — it will cross the ray at a point. You will mark that crossing as B in the next step.
• Place point B exactly where the arc crosses ray OA. B is the anchor for the next arc — it must sit right on the intersection.
• Arc tool: click centre B. The radius r from step 4 is locked. The arc swings up and crosses the O-arc at a point — you will mark that crossing as C in the next step.
• Place point C exactly where the O-arc and the B-arc cross each other. C is the key point — the ray through O and C will give the 60° angle.
• Ray tool: click O, then click C. This draws ray OD — the second arm of the angle. Because OB = OC = BC (all equal to radius r), triangle OBC is equilateral, so angle DOA = 60°.

The proof, step by step

Prove that the constructed angle ∠DOA equals 60°.

• By construction, OB and OC are radii of the same arc, so OB = OC.
• Also, BC is a radius of the second arc drawn from B, so BC = OB.
• Thus, OB = OC = BC, making triangle OBC an equilateral triangle.
• In an equilateral triangle, all interior angles are 60°. Hence, ∠DOA = 60°.

Worked example

In the construction of a 60° angle, an arc of radius 6 cm is drawn from point O to cut the base line at B. Another arc of the same radius is drawn from B to intersect the first arc at C. What is the length of the chord BC?

Since OB and BC are both radii of arcs with the same size (6 cm), and OC is also a radius of the first arc, triangle OBC is equilateral. Therefore, BC = OB = 6 cm.

• 3 cm
• 6 cm — correct
• 9 cm
• 12 cm