Triangle to Square

by Serhiy Grabarchuk Jr

Solution Step 0/0
  • Initial placement. Draw a full triangle outline.

  • Change 1. Move the triangle ruler up along the right side of the triangle just drawn and draw a segment along the ruler’s bottom side at some height from the triangle's base.

  • Change 2. Join D and B.

  • Change 3. Join C and E. The crossing is F.

  • Change 4. Draw a line through A and F in a way that the lower corner of the ruler lies on the triangle's base CB. The upper ruler's corner marks new point - G. The resulting GH equals the ruler's side and is perpendicular to CB.

  • Change 5. Put the ruler upside-down and side-to-side with the previously drawn triangle. Draw the new triangle along the ruler’s upper and left sides. A new point is created - I.

  • Change 6. Put the ruler so that its bottom side coincides with CB and simultaneously its lower right corner coincides with H. Draw the segment along the ruler’s bottom side. The ruler's lower left corner indicates a new point - J => JH = GH.

  • Change 7. Join I and J in a way that the ruler's lower corner is in J. Extend the segment till the upper ruler's corner. A new point is created - K => KJ is perpendicular to JH and KJ = JH = GH.

  • Change 8. Join K and G.

  • Result. The KGHJ is a square with all 4 corners as right angles and each side equaling the ruler's side.

Suppose you have a shape of an equilateral triangle with straight sides.

It is stated using just a pencil and the triangle as a ruler it’s possible to draw a regular square on a piece of paper.

How many times the ruler is required to change its position in order to achieve the goal, providing the ruler is placed on the paper already?


With the initial placement of the ruler on the paper it is required to change its position 8 more times to draw a regular square.

The step-by-step instructions are provided on the left.

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