Assignment 7
By: Olamide Alli
Let’s designate a few terms before we discuss the exploration, just to make sure that we are all on the same page. WE are going to call the circles big circle, little circle, and tangent circle. The little circle is inside of the big circle and we are trying to construct the tangent circle.
The tangent circle is going to be inscribed in the big circle because if we place the tangent circle outside of the big circle we will not be able to create a tangent to the little circle without intersecting the big circle. We also need to take note of the fact we can put the little circle anywhere in the big circle and have the potential to produce more than one tangent circle.
Let’s start the exploration with constructing the big circle and the little circle.
Draw a line from the center of the big circle and make sure it intersects the circumference of itself (the big circle). Make sure the line created doesn’t intersect the little circle.
Using the radius of the little circle and any point on the big circle, we can construct a circle surrounding the center on the big circle.
Now we must pick a point at the top of the newly constructed circle, call it small circle. In order to produce this point, extend a line through the center of the big circle that is also going to pass through the small circle. Now construct a line segment from the center of the little circle to the point on the small circle.
Now let’s locate the midpoint of the line segment that we made between the little circle and the small circle. We will now construct the perpendicular line through the midpoint, better known as the perpendicular bisector.
Let’s bring in the tangent circle. The tangent circle is going to be equidistant from the midpoint and the center of the little circle. There is a point of intersection of the perpendicular bisector of the line the joins the midpoint, the center of the little circle, and the line we originally constructed from the center or the big circle to the circumference. This point of intersection is the center of the tangent circle we have been trying to create!
Now let’s look at something else. Let’s construct a segment that connects the center of the little circle and the tangent circle. This new segment is the sum of the radius of the tangent circle and the little circle. This is what the image looks like.
Because we constructed the small circle from the little circle on the circumference from the big circle, the segment from the center of the big circle to the small circle is the same length. Now we have an isosceles triangle.
Let’s explore just a little bit further. Look at the locus of the circles that are tangent to the big and little circle. We can animate the circle and trace the locus of the center using the trace tool in Geometers Sketch Pad (GSP).
Please click the link to the GSP file to discover what the locus of the center of all tangent circles will be…don’t forget to click the button that says “animate point.”
GSP link
What did you see? Hopefully an….Ellipse!
Now what is going to happen when the little circle is put on the outside of the big circle? You will still get an ellipse. Did you notice that the distance between the foci is getting larger and larger?
What did you observe? I willing to guess the as the distance of the foci gets larger, and when move the little circle outside of the big circle that the locus of the centers will create a hyperbola. Click the GSP file to confirm the conjecture.
