There is quite a bit to think about when designing an irrigation system. However, many of my original conjectures about constructing circles inside squares would not be and efficient way to irrigate from a commercial standpoint. However, these constructions do seem to have more relevance on a residential level, even though crop farmers would most likely agree that much water is wasted by the way residential irrigation systems are designed. Regardless of these conclusions, the original geometry investigations are, I believe, still interesting. Thus, I will continue with the development of these constructions.

Before advancing to the construction of two circles in a square, let's first try to construct a single circle inside a square. This circle is considered inscribed.

Construction of Two Inscribed Circles

The first construction deals with two circles inside of a square, where each circle is tangent to two adjacent sides as well as tangent to each other. The challenge to this construction laid in trying to devise a way to keep both circles inside the square at all times. At first attempt, neither circle stayed inside of the square because the circles were constructed with their centers already defined on the diagonals. Then I was able to get the red circle fixed inside the square by realizing that the circle's greatest diameter would be 1/2 the length of the side of the square. By constructing the midpoint of one side (point O) and then constructing an arbitrary point, H, on segment OA, I was able to construct a circle tangent to H where the circle's boundaries were the four sides of the square.

The difficult part was making the green circle stay inside the square and remain tangent to two adjacent sides of the square as well as the red circle. In the figure below I noticed two things. First, that the point of tangency between the two circles was going to lie on the diagonal of the square. Second, I noticed that an isosceles triangle was formed by the three points of tangency (to be proven later). This meant that the diagonal of the square was also the perpendicular bisector of the base of the isosceles triangle and it was the angle bisector of the opposite angle. Since the circles are to be tangent, the three points of tangency of the green circle would also form an isosceles triangle with the same properties holding as the other isosceles triangle. If I can construct one of the points of tangency to the square then I can construct the green circle. The measure of the opposite angle is 45 degrees, so half of that is 22.5 degrees. Since I know that the diagonal is also the angle bisector of the isosceles triangle of the green circle, if I rotate this part of the diagonal 22.5 degrees it will intersect one of the necessary sides at the targeted point of tangency.

You can click here to see the GSP file. See what happens to the green circle when you move point H around. What do you notice about the green circle when H approaches point A? You will notice that the green circle disappears as H approaches A. Is there a way to make this construction so that will not happen? The answer to this question is yes and can be found in the feature construction link.

Point H is located on segment OA, which is half the distance of one side of the square. Click here to move point H on OA to see how the red and green circles are related.

Both the blue and yellow circles have equal area regardless of their sizes. Click here to see how much area is actually used when the circles' areas are changed.

What about when there are four circles inside the square?

Continue to the Geometric Proofs Page

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