Before we can discuss the construction of an inscribed cricle, we need to make sure that we know what an inscribed circle is! Click on the link for "inscribed circle" to get the definition and an illustration of what an inscribed circle looks like.
Since we these constructions are in keeping with irrigation farming we will work with one of the major surface areas for irrigation farming, the square. Recall that this page is devoted to discussing the construction of an inscribed circle. However, you will need to understand the construction of a square as well. If you're set on the construction of the square, let's get started on our inscribed circle!
Let's start off with Square ABCD:
Keep in mind, the circle is only inscribed if it is, in fact, inside the square. Sorry guys........no "make square, make circle and stick circle inside". That's not taking into consideration the properties that make a circle inscribed. Since the circle needs to be tangent to all for sides we need to construct exactly one point of intersection between the circle and the square on each side of the square. Hmmm, this seems a daunting task at first, but if you're familiar with what "tangent" means, this should give you some clues. If there's exactly one point of intersection on each side and each side is equal, chances are that the point of tangency will be in the same place as well. In fact it is! So, where might the point be and still keep our circle inside the triangle? Well if we pick a point on each side that is close to the vertices, then our circle will be too big. But if we pick a point that is as far away from any given vertex on any given side, then we will be in good shape.
Do you know what that farthest point from any given vertex is called? You guessed right, it's the midpoint of each side! So, let's go ahead, as our next step, and construct the midpoint of each side and call them W, X, Y, and Z. Now, let's work on constructing the centerpoint of the square. Why you ask? Well, if we can find the centerpoint of the square, we have also constructed the centerpoint of the inscribed circle. Yes, the two centerpoints are coincident.
Since there is more than one way to construct the center of a square we will use the method related to points of tangency. When finding a point of tangency, between a straight edge and a curve, one should consider the property of perpendicularity. If we construct segment XZ, for example, it will be perpendicular to segment DB and AC. Why you ask? Recall that Z and X are equidistant from segments AB and CD. So, by definition of parallel, segment XZ will be parallel to AB and CD. Thus, XZ will be perpendicular to BD and AC and segment WY will be perpendicular to AB and CD as defined by the square. So constructing XZ and WY its point of intersection will give us our centerpoint, P.
Looking at point P carefully, you will see that it is equidistant from points W, X, Y, and Z. By definition of a circle, W, X, Y, and Z are radii of a circle through centerpoint, P. Thus, by constructing a circle with centerpoint P and any one of the four midpoints of Square ABCD, we have our inscribed circle. This construction works because the circle is totally based upon the given square. Therefore, if you check in GSP and manipulate the vertices of the square, the circle will remain inside the square and tangent to all four sides at points W, X, Y, and Z. To check the GSP construction, click on the finished construction below.
Now that you know how to construct an inscribed circle, try your hand at constructing two inscribed circles!
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