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 circlesbigcircle,littlecircle, andtangentcircle. Thelittlecircle is inside of thebigcircle and we are trying to construct thetangentcircle.

Thetangentcircle is going to be inscribed in thebigcircle because if we place thetangentcircle outside of thebigcircle we will not be able to create atangentto thelittlecircle without intersecting thebigcircle. We also need to take note of the fact we can put thelittlecircle anywhere in thebigcircle and have the potential to produce more than onetangentcircle.

Let’s start the exploration with constructing thebigcircle and thelittlecircle.

Draw a line from the center of thebigcircle and make sure it intersects the circumference of itself (thebigcircle). Make sure the line created doesn’t intersect thelittlecircle.

Using the radius of thelittlecircle and any point on thebigcircle, we can construct a circle surrounding the center on thebigcircle.

Now we must pick a point at the top of the newly constructed circle, call itsmallcircle. In order to produce this point, extend a line through the center of the big circle that is also going to pass through thesmallcircle. Now construct a line segment from the center of thelittlecircle to the point on thesmallcircle.

Now let’s locate the midpoint of the line segment that we made between thelittlecircle and thesmallcircle. We will now construct the perpendicular line through the midpoint, better known as the perpendicular bisector.

Let’s bring in thetangentcircle. Thetangentcircle is going to be equidistant from the midpoint and the center of thelittlecircle. There is a point of intersection of the perpendicular bisector of the line the joins the midpoint, the center of thelittlecircle, and the line we originally constructed from the center or thebigcircle to the circumference. This point of intersection is the center of thetangentcircle we have been trying to create!

Now let’s look at something else. Let’s construct a segment that connects the center of thelittlecircle and thetangentcircle. This new segment is the sum of the radius of thetangentcircle and thelittlecircle. 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 thelittlecircle is put on the outside of thebigcircle? 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.