Construct an equilateral triangle inscribed in a circle

Key Notes:

An equilateral triangle has all three sides congruent and all angle measures equal to 60°.

Two chords in a circle are congruent if and only if their corresponding central angles are congruent.

Learn with an example

The diagram below shows a nearly completed construction of an equilateral triangle inscribed in ⨀A with a vertex at B. Complete the construction.

Part of the construction was done for you. Here are the steps to create this part of the construction.

Start with the objects in the diagram below.

  • Draw the line through A and B.
  • Draw a circle with radius AC centred at C.
  • Mark the points where ⨀A and ⨀C intersect. Call them D and E.
  • Since C, D, and E are all on ⨀A, AC=AD=AE. Since A, D, and E are all on ⨀C, AC=CD=CE. This means that AC=AD=AE=CD=CE.
  • So, △ACD and △ACE are congruent equilateral triangles.
  • Draw the segment between B and D.
  • Draw the segment between D and E.
  • Complete the construction.
  • To complete the construction of an equilateral triangle inscribed in ⨀A with a vertex at B, carry out the following step:
  • Draw the segment between B and E.
  • Since △ACD and △ACE are both equilateral, m∠CAD and m∠CAE are both 60°. So, the supplementary angles ∠BAD and ∠BAE are both 120°.
  • Also, m∠DAE = m∠CAD + m∠CAE = 60° + 60° = 120°. Since the central angles ∠BAD, ∠BAE, and ∠DAE are all congruent, so are the corresponding chords BD, BE, and DE.
  • So, △BDE is an equilateral triangle.

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