Inscribed Circles of a Semicircle
by
Ángel M. Carreras Jusino
The purpose of this essay is to explore inscribed circles of a semicircle. Lets begin by defining what is an inscribed circle. An inscribed circle of a polygon is defined as the circle that is tangent to each side of the polygon. But what about an inscribed circle of a semicircle? No formal definition for this term was found in textbooks or internet sites. For the purpose of this essay I defined the inscribed circle of a semicircle as any circle that is tangent to the arc and the diameter that forms the semicircle.
Let’s first take a look at the biggest inscribed circle of a semicircle. Intuitively we can say that this circle is going to be tangent to the center of the semicircle, its radius length is going to be half of the radius of the semicircle (i.e., its diameter length is the radius length of the semicircle) , and that its center would lie in the midpoint of the radius of the semicircle which is perpendicular to the diameter that forms the semicircle (see Figure 1).
Figure 1.
I claim that any other inscribed circle would be smaller that the aforementioned, later in this essay will be presented an argument for this.
Now let explore other inscribed circles (see Figure 2).
Figure 2.
Note that the center of this circle lies in a radius of the semicircle. Particularly, each radius of the semicircle contains the center of one inscribed circle. Based on this fact is proposed to define relationships between each radius of the semicircle and characteristics of the inscribed circle whose center lies on it.
Each radius of the semicircle could be identified by the angle (θ) that it forms with the left side of the diameter (from here on "left radius") of the semicircle (see Figure 3).
Figure 3.
Two pieces of information of the inscribed circle that would be interesting to investigate are the radius of each inscribed circle and the position of its center.
Let's focus first in the radius of each inscribed circle. For every inscribed circle (except the ones that are at 0, π/2, and π) a right triangle, like the shown in Figure 4, can be created.
Figure 4.
Now, let's take a closer look to this triangle.
Figure 5.
(Note: For convenience we will refer to the radius of the semicircle as R and to the radius of a inscribed circle as r.)
Considering the small triangle the length hypotenuse can be written as R - r.
Figure 6.
Using trigonometry a function for r in terms of θ can be written.
This formula provides the length of the radius for every inscribed circle given the angle of the radius of the semicircle in which its center lies with the left radius.
Now, let's see its graph in the Cartesian plane. (Note: For graphing purposes R = 1 was used as the radius of the semicircle in all the graphs with coordinate axes.)
Figure 7.
From this graph can be observed that the inscribed circle with the greatest radius (r = ½R) occurs at π/2, as conjectured at the beginning of this essay, and no circles (r = 0) at the endpoints of the diameter of the semicircle (θ = 0, π) .Now that the radius of every inscribed circle can be determined based on the angle formed between the radius in which the center of the semicircle lies and the left radius, let's look for the position of the center.
To do this let's consider that the center of the semicircle lies in the origin of the Cartesian plane and its diameter lies in the x-axis.
Figure 8.
Let's look again to the right triangle but now in terms of Cartesian coordinates.
Figure 9.
First note that the length of the radius for each inscribed circle is the y-coordinate of the center of the inscribed circle.
Applying the Pythagorean Theorem:
Surprisingly, the locus of the center of the inscribed circles in a semicircle is a vertical parabola with domain [-R, R], range [0, R/2], vertex (0, R/2), focus (0, 0), and directrix y = R.
Now, with all this information is possible to construct the family of inscribed circles of any semicircle. As shown in the following animation.
Also, because is now known that the locus of the center of the inscribed circles is a parabola a construction with interactive geometry software, as Geometer's Sketchpad (GSP), is possible. A GSP script tool that constructs the locus of the centers of the inscribed circles and the inscribed circles given the endpoints of the diameter of the semicircle is provided here (additional details of how to manipulate the tool are provided in the file).
Using an approach similar to the used in the script tool, an applet was created with JavaSketchpad (see below). This applet allows the user to explore the inscribed circles of the semicircle as the center of the inscribed circle is moved or the size of the semicircle is changed. (Note: Since JavaSketchpad does not allow the construction of semicircles (arcs), in the applet is presented a circle instead of a semicircle).
