SOME NON-ELEMENTARY PROBLEMS ABOUT CIRCLES.

Raimond A. Struble.

Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh
Raleigh, NC.
1/19/2006.

Send comments and correspondence to: raimondstruble@yahoo.com


INTRODUCTION



In [1], we consider pictures illustrating the resolution of an elementary problem about circles which involve the configurations manifested by three mutually tangent circles. In this brief companion note, we reconsider these configurations (triple-spiked regions), along with some rather non-elementary geometric propositions of independent interest which concern the circles. The simple statements of the propositions seem elementary enough, it is only the proofs (postponed to the last section) that are non-elementary. Believing the propositions to be novel, we explore a few manifestly certain consequences and speculations concerning these circles.

Despite the title and a virtual gallery of pictures depicting circles, the announcement and proof of a new theorem about triangles is the principal objective of this note.

THE PROPOSITIONS



What we claim is that for any three circles, non-intersecting, tangent to one another:
(i) the three line-segments drawn from the centers to the opposing tangency-points meet at a common point;

(ii) the three tangency-lines meet at a common point;

and (iii) these two meeting-points are coincident only if the three circles possess a common radius. (Only then are they coincident with the center of the largest circle that can be embedded within the triple-spiked region which they form.)


Picture 1:

887.

PLAUSIBILITY ILLUSTRATIONS

Consider the family of all circles centered at the center of the largest of the three tangent circles.

Concerning (i), Picture 2 illustrates that the crossing of segments Aα and Bβ occurs to the left of PCL for "smaller" circles, and to the right of PCL for "larger" circles (relative to the given largest circle).

Picture 2:

928.
Therefore, one surmises that there is a meeting-point somewhere on PCL.

Concerning (ii), Picture 3 illustrates that the crossing of the tangents at α and β occurs to the right of VTL for "smaller" circles, and to the left of VTL for "larger" circles.

Picture 3:

923.
Therefore, one surmises that there is a meeting-point somewhere on VTL.

Concerning (iii), these three pictures, along with Pictures 4 and 5, actually validate (iii), because of the symmetry (or lack thereof) involved.

Picture 4:

890.


Picture 5:

891.


It is suggestive that the above arguments could be rendered rigorous using the asymmetry displayed by the smaller A and B circles. On the other hand, if circles A and B possess a common radius, then (i) and (ii) are clearly valid (again because of the symmetry involved).

OTHER CONFIGURATIONS

Appropriate versions of Propositions (i) and (ii) evidently hold for still other fascinating configurations of three mutually tangent circles, as illustrated in Picture 6, when one circle contains one or both of the other two.

Picture 6:

916.


Here again, because of the symmetry this is certainly the case if two internal circles possess a common radius. Also, appropriate versions of them hold if there is a triple tangency-point.

THE CONSEQUENCES.

It is intriguing to imagine an animated-type program, showing how the three meeting points separate in the simple Picture 5, morphing into the complicated Picture 4 and numerous others (such as in Picture 7), as the sizes of the circles evolve.

Picture 7:

910.


Imagine the intriguing kaleidescope if such an animated program with colors were used to demonstrate the successive filling-process of the triple-spiked region considered in [1], with this extra framework. The latter would come to life in dramatic fashion indeed, as the unfilled regions would begin to disappear, and the filling circles complete their mission.

As demonstrated in Picture 8, the area of the triple-spiked region can be determined as the difference between the area of triangle ABC and the combined area of six circular sectors within the interior triangles ABD, BCD and ACD.

Picture 8:

911.




Alternatively, as demonstrated in Picture 9, the area of the triple-spiked region can be determined as the difference between the area of triangle ABC and the combined area of six circular sectors within the interior triangles ABE, BCE and ACE.

Picture 9:

912.


Of course, there is another (certain) way to determine the area of the triple-spiked region using the imbedded circle. (See Picture 10).

Picture 10:

913.


First, the radius r of this circle is determined from triangle ABC, where the three interior triangles ABF, BCF and ACF involve radial segments of the embedded circle. The area of the triple-spiked region is then determined as the difference between the area of triangle ABC and the combined areas of six circular sectors of these last interior triangles.

The connections between the three methods for determining the areas of the triple-spiked regions might be clues to rigorous (geometric) proofs of the propositions themselves.

In [1], it is shown that the ratio of the area of the imbedded circle to that of the triple-spiked region tends from a likely maximum of 0.466 (with circles of equal radii) to zero when a small circle is squeezed in between two much larger ones. (See Picture 7).

Here, the ratio of the area of the inferred triangle DEF to that of the triple-spiked region becomes some sort of measure of the asymmetry present, clearly zero when two or three of the circles possess a common radius but otherwise positive, and certainly bounded by the number 1, since the triangle always lies within the region. The three points D, E and F cannot become collinear if the three circles possess unequal radii.

Picture 11:

914.


The triangle DEF itself is barely visible in most graphical displays, but can be seen if the circles are magnified and the radii are appreciably different from one another. Picture 11 reveals that with a very large circle at C, the difference between the other two radii is what forces triangle DEF to "open up". The principal consequences of this difference is to shift vertex E to the left (toward the larger of circles A and B) away from edge DF, which ensures that the area is indeed positive. The center F of the embedded circle also shifts to the left, but necessarily less so than E. (See also Picture 4).

It is not difficult to see from Picture 12 that the triangle edge DE becomes perpendicular to the centers' line BC when a small circle is squeezed in between two much larger ones. It is, of course, always perpendicular to BC should the larger circles possess a common radius, for then the points D, E and F do become collinear.

Picture 12:

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Speculating, we wonder if the area of the triangle DEF could possibly be determined from geometric considerations of the three sets of interior triangles, and just what its maximum fractional value could be? (surely much less than 1 or even 0.466)

We note in passing that our various pictures display the only possible configurations of either three or four mutually tangent circles. There are none for larger numbers of circles. The celebrated problem concerning eight instances of a circle that is tangent to three others, involves intersecting circles which are not mutually tangent to one another (except for the configurations used here).

ANOTHER PROPOSITION



It is a very simple observation (as in Picture 12) to note that the centers of any three mutually tangent circles produce a triangle whose edges display the tangency-points. It is not so simple to observe the opposite truth.
Proposition (iv): the edges of any triangle display the tangency-points of three mutually tangent circles with centers located at the vertices.
Imagining still another animated-type program showing increasing and decreasing circles with centers located at the vertices so as to produce the tangency-points, is certainly one intuitive way to establish this proposition. But a much more constructive technique for such a program is to employ a modified form of Picture 2. We replace PCL with the line-segment drawn from C that actually passes through the crossing-point. Then since Proposition (i) holds, the intersection of this line-segment with the edge AB yields a prospective tangency-point from which circular arcs about A and B can be drawn. The other tangency-points become evident as the circle at C enlarges and eventually encounters these arcs at the correct α and β locations. An even more constructive technique is to use not only this modified Picture 2, but also a similarly modified Picture 3. We replace VTL with the line segment passing through the crossing-point which is perpendicular to the edge AB. Then since Proposition (ii) holds, the latter intersection yields another prospective tangency point on AB. The midpoint between these two prospective tangency points yields a much improved prospect with which to continue the search. The tangency points can be obtained constructively using only the interior lines drawn from the centers, together with perpendiculars from the edges, and linear measurements made along the edges. (The circles merely enhance the panorama displaying the triple-spiked regions.) Since iterations of this dual technique do indeed converge to the exact tangency point on AB, then the graphic scheme amounts to a constructive validation of Proposition (iv).

Proposition (iv) provides for a-priori designs of triple-spiked regions through the mechanism of triangle selections. For example, if the triangle is very obtuse, then the resulting region will reflect the squeezing of a small circle between two large ones, as seen in Picture 12. While at the other extreme, an acute, spire-like triangle will reflect the expansion of one triangle beyond the other two, as in Picture 8. Of course, isosceles and equilateral triangles reflect clearly identifiable symmetrical regions. Also, it is obvious that the areas of the triangles will reflect those of the triple-spiked regions, but not in any clearly identifiable fashion. Certainly, a precise formulation would be of some interest (at least if reasonably simple). Interestingly enough, the areas of the triangles cannot possibly reflect those of the embedded circles, because of the aforementioned cases of obtuse triangles (their fractional values approach zero).

THE TRIANGLE THEOREM.



In conclusion, we observe that because of Proposition (iv), the original propositions can be rephrased as a single, somewhat mysterious existence theorem about triangles.

TRIANGLE THEOREM. Let a triangle have vertices at points A, B, and C.
(i). There exists a unique interior point, D, for which the three line segments emanating from the vertices and passing through D intercept the edges of the triangle at three opposing points, a, b, and c, SATISFYING length equalities Ab=Ac, Ba=Bc, and Ca=Cb.

(ii). There exists a unique interior point E, for which three line segments emanating from E to the points a, b, and c, are perpendicular to the edges of the triangle.

(iii). There exists a unique interior point F, for which there exists a positive number r, and for which three line segments emanating from F to the vertices have lengths, WHEN SHORTENED BY r, given by Ab, Ba, and Ca.

(iv). The interior points D, E, and F are coincident only for equilateral triangles.


Picture 13:

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PROOF OF THETRIANGLE THEOREM. This is a very unusual type of proof for a geometrical theorem, using analytical differentiation based upon pictures. For Part (i), we examine Picture 14, where initially the crossing-point D of segments Bb and Cc falls on the segment A0 according to the obvious truth of the theorem for isosceles triangles. For a small angle, θ, the resulting crossing-point Dθ of the segments Bθbθ and Cθcθ falls on segment AθDθ, which when extended to edge BθCθ, intercepts approximately at Aθ. Clearly, the error Eθ in this location (if not zero) is infinitesimally small in comparison to θ. So we conclude that dEθ/dθ=0, for θ=0.

Picture 14:

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But this argument can be made for any other (permissible) θ-value. In fact, if, as in Picture 15, the segment AθDθ leads to an error Eθ in the location of Aθ, then the incremental change to the error, Eθ+Δθ, corresponding to an incremental change, Δθ in θ is, similarly, infinitesimally small in comparison to Δθ

Picture 15:

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So we conclude that dEθ/dθ=0, for all θ. The changes are significantly retarded relative to those of the triangles itself. This interval framework is, in fact, stable under changes in θ. Since Eθ is continuously differentiable and Eθ=0 for θ=0, it follows that Eθ=0 for all θ, which proves (i).

Part (ii) of the theorem can be proved using the same type of analytical arguments. Parts (iii) and (iv). have already been established in the consequences and illustration sections.

A standard geometrical proof of this theorem is very desirable, but seems to be very difficult.
H. REFERENCES.


1. Struble RA.
An elementary problem about circles. Infinite products, filling programs, and integration.
GALLERY.



887.          888.          889.          890.
891.          892.          893.          894.
895.          896.          897.          898.
899.          900.          901.          902.
903.          904.          905.          906.
907.          908.          909.          910.
911.          912.          913.          914.
915.          916.          917.          918.
919.          920.          921.          922.
923.          924.          925.          926.
927.          928.          929.          930.
931.          932.          933.          934.
935.

H. REFERENCES.


1. Struble RA.
An elementary problem about circles. Infinite products, filling programs, and integration.

2. Mikusiński J.
The Bochner Integral.
New York, San Francisco: Academic Press. Harcourt Brace Jovanovich, Publishers. 1978;:. Pure and Applied Mathematics.
Basel: Birkhäuser. Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften: Mathematische Reihe. [Textbooks and monographs from the area of exact sciences: mathematical series.] 1978;55:.
ISBN: 3764308656, 233 pages.

3. Struble RA.
Infinite Products Rescued.
http://www.infiniteproduct.info/struifpr.htm

4. Struble RA.
Infinite Products and Integration.
http://www.infiniteproduct.info/struitgr.htm

5. Derbyshire J.
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.
New York: Plume Books. 2004. ISBN: 0452285259.


Last updated: 1/19/2006, by Raimond A. Struble, PhD.