The Department of Mathematics Education
Write Up #7
Basic Construction Concepts
Tangent circle construction in general is based on the idea that, given two circles, a and b, the circle tangent to both will have a center lying along the radius of one circle, say a, and one point of tangency, M, of the new tangent circle will coincide with the intersection of the radius of a with circle a:
Note that the circle in the dotted lines is assumed to exist at this point – it is yet to be constructed and is only shown as a guide for reasoning through the required construction steps.
Next realize that the distance between the center of b and the center, T, of the supposed tangent circle is the sum of the radii of b and the new tangent circle:
Imagining that this is the leg of an isosceles triangle, an equal leg of this triangle can be found along TM by adding a radius of b centered at point M:
At this stage of the construction it should be clear that the center, T, of the tangent circle will be located along the perpendicular bisector of the base of this isosceles triangle. Also since we already know T lies along the radius of a, point T is at the intersection of the perpendicular bisector and the radius of a:
Note that the selection of the names point M on the large circle and point T on the newly created tangent circle are for a reason. Point M implies “Motion” and point T implies “Trace”. Scripts for this construction, as well as four others, can be found in the Geometer SketchPad ver. 4.04 file Asmt7_WGD.gsp. The construction above is found on page one of this file, but it also contains scripts for this and the other constructions that, if this file is place in the GSP Tool Folder, may be used to quickly construct a number of tangent circles.
Back to the “Motion” and “Trace” points, once the script for this circle is used, select only the point M and animate. The point T has already been selected as a trace point in the script. Depending on whether circle b is inside or outside circle a, the center of the tangent circle traces out either an ellipse of a hyperbola:
By a similar reasoning process, a number of other approaches to constructing the tangent circle may be taken. These, along with their scripts are found in the file Asmt7_WGD.gsp. The first four variations, which includes the approach discussed above, may have the initial point, M, located on the large circle or on the small circle. The small circle may be located either inside the tangent circle are outside:
The fifth construction considers that case where the large circle, a, has an infinite radius; in other words it is a line:
Exploration the Traces of Triangle Centers
There is a myriad of curiosities contained in this construction, such as the formation of conic sections for the chosen trace and motion points, such ellipses and hyperbolas, whether one of the construction circles is on the interior or exterior of the other construction circle. The fifth case traces out a parabola. In the process of playing with the tangent circle, I became curious as to whether the points on the Euler line contained within the isosceles triangle used in the construction offered any possibilities for further exploration. This offers literally hundreds of avenues to explore, so I began with the first construction one and continued until I discovered a relationship that particularly piqued my curiosity.
Starting with the first construction (WGD 7-1 Motion Point, M, on Large, Small Outside T Circle). I explored the traces of the triangle centers, G, C, H and I as the point M moved on its circular path. The centroid G was modestly interesting in that it form elliptical and hyperbolic paths, similar to the center of the tangent circle.
The incenter, I, of the triangle was similar to the centroid with the exception that when circle b is exterior to a, the incenter for a line at the base of the hyperbola traced by the tangent circle center.
H was of somewhat greater interest in that is forms a what appears to be a perfectly linear path
Finally, the trace of the circumcenter, C was considered:
Of course these same exercise could be done on the remaining four constructions. But in terms of discussing the behavior of the triangle centers in depth, the discussion soon becomes intractable given the number of possibilities. So how do we chose a feature deserving greater consideration? Although this is obviously a very subjective question, feature that I find of the greatest intrigue are those that exhibit great constancy in the fact of radical change of the environment. Notice that the all of the triangle centers exhibit a change in their limiting behavior whether circles a and b occupy the same region or a disjoint region; that is, of course, except for the circumcenter. It appears to follow a linear path as point M traces circle a regardless of whether b occupies the same region as a. Why should this be so?
Trace of the Circumcenter
That the trace of the circumcenter should follow a linear path is actually nothing too earth-shaking when one recalls the definition of the circumcenter versus the definition of the triangle connecting the centers of the centers of the tangent circles. Considering each case separately, the underlying reason for this behavior is the same. This being the case, only the case for the ellipse is considered.
Take first the case of circle a lying inside circle b. Notice that as point M moves along circle a, that centers of a and b are stationary and connected by one leg of the triangle used in the construction of the triangle circumcenter. This single fact, while not explaining yet the motion of C, does explain the path that C must be restricted to. Recall that the definition of the circumcenter is the intersection of perpendicular bisectors of the triangle sides. One of these sides is the segment connecting the stationary foci of the ellipse traced by C. Therefore, the circumcenter will not exist outside the path of this perpendicular bisector of the major axis of the ellipse. Regarding the motion of C, this is determined by the position one (actually both) of the other bisectors, which is not stationary. As T traces out the ellipse, note that at time M lies along the line through the major axis of the ellipse. This is when C is at infinity because, the triangles acute angles are zero and the perpendicular bisectors of the triangle side are parallel. Therefore they never meet form the circumcenter. But as the acute angles leave zero, that is M is on one side or the other of line through the major axis, the circumcenter approaches from “+” or “-“ infinity:
Note that this approach from + and – infinity occurs twice, each time M crosses the major axis of the ellipse. Despite the differences between the ellipse and the hyperbola form by the center of the tangent circle, the line connecting the foci of the two curves is stationary. Therefore the consideration for the locus of points of the circumcenter is the same for both curves.