A typical geometry question asks whether a given curve is inscribed in a square. To find the answer, you sketch a figure with sides of equal length and calculate its area. Then, you divide the figure into two right triangles with legs (or hypotenuses) that sit hypotenuse-to-hypotenuse. The sum of the area of these two triangles divided by their number is equal to the circle’s radius times 2. This provides a convenient numerical value for the inscribed-square’s perimeter and side lengths.
For many students, this kind of problem seems to be beyond them. But, with a little guidance, they can tackle it. The article, Which one of these is not a step used when constructing an inscribed square using?, is designed to give students the practice they need. It also shows how to use the perpendicular bisector of a diameter and a compass to construct the four vertices that define an inscribed square in a circle.
The article is accompanied by an animation that guides students through the steps. It is also available as a printable step-by-step instruction sheet, which can be useful for making handouts or when a computer is not available.
The question is easy to state and has a nice geometric flavor (with a touch of topology for interest). However, it evaded a complete solution until 1989 when Stromquist proved that every local monotone plane simple curve admits an inscribed square. A local monotone curve is a curve that can be locally represented as the graph of a function y = f(x).