OBSERVATIONS CONCERNING
MUTUALLY TANGENT CIRCLES,
TRIANGLES, AND CASCADES.
DRAFT COPY ONLY.
8/30/2008.
Raimond A. Struble.
Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh
Raleigh, NC.
http://www.infiniteproduct.info/struobsv.htm
Send comments and correspondence to: raimondstruble@yahoo.com
ABSTRACT
This work is a comprehensive study of the associated geometry surrounding
(non-intersecting) externally and internally, mutually tangent circles, using
analytical and especially pictorial observations. The inevitable synthetic
connections with triangles is afforded a novel treatment, motivated by
alternative, necessary arguments, created by the internal tangency
situations. Cascades of triangles based upon (external) Gergonne
concurrencies (using alternative terminology) are shown to be topped off
by obtuse or right-angled triangles, with families of descending
acute triangles collapsing down upon the limit-points of the corresponding
Gergonne points. In this study, two new (relevant, but unproved) synthetic
concurrency problems about rectangles are encountered, and displayed
as serious challenges for the more inquisitive readers. (It is conjectured
that proofs of these problems are not amenable to the use of Cevas' theorem.)
A three-dimensional view of the associated spheres phenomena is described
in some detail. This view reveals that the embedded sphere (circle) should
be considered the primary item when dealing with mutual tangency
configurations. Cascades based upon internal and external tangencies are
shown to have bottom acute or right-angled triangles, with ascending families
of obtuse triangles, that tend upward to infinity. Natural cascades,
consisting of descending and ascending circles and triangles are revealed,
and are then offered as models for real-life applications.
INTRODUCTION
In [1], we consider pictures illustrating the resolution of an
elementary geometric problem about circles that involves the configurations
manifested by three mutually tangent circles. In this extended work, we
reconsider these configurations (triple-spiked regions), along with some
elementary geometric propositions of independent interest that intimately
concern the circles. It seems appropriate then to explore a few obvious
consequences, and to speculate upon others motivated by the revealing
pictures. We emphasize that our main concern here is with observations
leading to discoveries, not rigorous arguments substantiating
the discoveries. Readers may wish to attack the more elusive observations.
Frequently, we shall call upon the imagination of the readers for further
discoveries. Knowledgeable readers may enjoy an analytic (non-Ceva) proof
of the Gergonne concurrency, as well as the challenge of two new concurrency
problems for rectangles, all presented in later chapters, along with
other real-life vistas
[7,8].
After presenting numerous pictures depicting circles, a theorem about
triangles (as it relates to the mutually tangent circles) brings our
observations into focus with traditional concerns about points, lines, and
triangles. This perspective, however, is definitely not the traditional one.
For example, we purposely ignore the incircle and Gergonne terminology.
Some fascinating new geometry, reflected in cascades of triangles
is presented, for which the tops of the cascades are found to be (precisely)
obtuse or right-angled triangles. These triangle-cascades lead to a greatly
enlarged view of the geometry of points, lines, and triangles.
A three-dimensional view further clarifies all the observations in a more
natural setting, which is something of an innovation, inasmuch as synthetic
geometry of this sort has focused mainly upon two-dimensional configurations
[3,4]. Our explorations begin with the examination of some motivating
pictures, and conclude with fascinating pictures of ascending and descending
cascades of circles and triangles.
CHAPTER 1.
THE CIRCLE PROPOSITIONS.
What one observes in Figure 1, and can prove,
is that for any three non-intersecting circles, 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 equal radii. Only then are they coincident with the
center of the largest circle that can be embedded within the triple-spiked
region which they form, as seen in Figure 2.
Figure 1:
2321.
Figure 2:
2322.
Also illustrated in Figure 1 is the very interesting observation
that the three rays passing through the centers intersect the three rays
passing through the tangency opints at three collinear points.
(One intersection is far removed to the right.) This is a recently
established fact (in Ref. 6 within a much broader context), but quite
auxiliary to the purposes of the current study.
ALTERNATIVE CONFIGURATIONS
Appropriate versions of Propositions (i), (ii) and (iii)
hold for still other fascinating configurations of three mutually tangent
circles, illustrated in Figure 3, in which one circle contains
one or both of the other two. Appropriate versions of Propositions
(i), (ii) and (iii) also hold if there is a multiple
tangency-point, as illustrated. We shall examine these alternative
configurations in detail at appropriate points of this study.
Figure 3:
2323.
Perhaps one should notice that here, the meeting-points fall outside of the
triangle determined by the centers of the circles. These pictures, as well as
the following ones, remind us of wheels, spokes, gears, and the frameworks
of velocipedes of the past, and of children's toys (Figure 3,
of possible visitors from outer space).
SOME CONSEQUENCES.
I. It is intriguing to imagine an animation program, showing
how the three meeting points separate in the simple Figure 2,
morphing into the complicated Figure 1, as well as numerous others
(such as in Figure 4), as the sizes of the circles evolve.
Note that one of the meeting-points can actually fall within
one of the circles. The other meeting-point, of course, cannot.
Figure 4:
2324.
II. Imagine further, in the alternative configurations
of Figure 3, some changing expressions on the faces of outer space
visitors (as the sizes of the circles evolve).
III. As seen in Figure 5, the area of the
triple-spiked region is determined as the difference between the area
of triangle ABC, and the combined areas of six circular sectors
within the interior triangles, ABD, BCD, and ACD.
Figure 5:
2325.
IV. As seen in Figure 6, the area of the
triple-spiked region is also determined as the difference
between the area of triangle ABC, and the combined areas
of six circular sectors within the interior triangles, ABE,
BCE, and ACE.
Figure 6:
2326.
V. Of course, there is still another (certain) way
of determining
the area of the triple-spiked region using the embedded circle, and its
center, F, which represents now a third meeting-point. (See
Figure 7, where it adds a pedal sprocket to our velocipede.)
Figure 7:
2327.
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 combined areas of the three interior triangles must be equal to
that of triangle ABC. 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.
Although straightforward in principle, any (exact) calculation
of the area of a triple-spiked region using these prescriptions becomes
a formidable task. But the most straightforward way, subtracting the combined
areas of the three original circular sectors from the area of triangle
ABC, is not a great deal simpler.
VI. Carefully drawn pictures suggest that the three
meeting-points, D, E and F might be collinear,
and that F is generally located approximately at the
mid-point between the other two (including the alternative configurations
in Figure 3.)
VII. It is not difficult to see from Figure 8
that the extension of the line segment DE becomes perpendicular
to the centers' line BC as a small circle is squeezed in between
two much larger circles. It is, of course, always perpendicular to BC
should the larger circles possess a common (numerical) radius. The points
D, E and F are then certainly collinear, and
Propositions (i), (ii) and (iii) clearly hold, but the
mid-point approximation ultimately breaks down completely, as the third
circle shrinks down to a point ([1], Figure 3).
Figure 8:
2328.
The (approximate) mid-point condition appears to hold only so long as the
meeting-point D remains within the embedded circle. This condition
requires that the ratio of the largest to the smallest radii be something
of the order of 5 or less. Exact values required could be calculated,
but the calculations become other formidable tasks. Some readers may wish to
attack the simpler concurrency problem, where the meeting-point D
passes into the interior of the smallest circle at A, while the other
two have equal radii. In a surprising coincidence, the requirement involves
the familiar 3 ×4×5 right triangle.
VIII. We note in passing that our various pictures (including
Figure 3) display the only possible configurations of either
three or four mutually tangent circles. There are none for larger numbers
of circles (except for multiple tangency points with nested circles).
The celebrated problem of Apollonius, concerning (possibly) eight instances
of a circle that is
required to be
tangent to three others, involves circles which
may intersect and are not mutually tangent to one another
(except for the configurations used here).
CHAPTER 2.
A TRIANGLE PROPOSITION.
It is a very simple observation (as illustrated in Figure 1
and 8) to note that the centers of any three mutually (externally)
tangent circles produce a triangle whose edges display the tangency-points.
It is not quite so simple to observe the opposite truth.
Proposition (iv): The edges of any triangle display the
tangency-points of three mutually (externally) tangent circles with centers
located at the vertices.
An intuitive way to envision this proposition is to imagine still another
animated-type program, showing increasing and decreasing circles with
centers located at the vertices so as to produce the tangency-points.
This program can be made a little more precise by considering various-sized
circles centered at angle A in Figure 8 (the largest angle of
the triangle). If too large, then the corresponding tangent circles, centered
at B and C do not meet; while if too small, then the
small one does not meet the other two. The tangency point along BC
is thus obtained from the correct size of the circle at A,
which then determines the other tangency points.
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
Figure 8. While at the other extreme, an acute, spire-like triangle
will reflect the expansion of one circle well beyond the other two.
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). For an equilateral triangle,
the ratio of the two areas is precisely
[1 - π/2 √3] ~ 0.0931, and
for a 1 × 1/√3 obtuse isosceles triangle, precisely
[1 - π/2√3] - π(2 - √3)2/33/2
~ 0.0496 ([1], page ?). The ratio for the
3 × 4 × 5 right triangle is something like 0.078.
Perhaps approximations might possibly prove useful.
CHAPTER 3.
THE TRIANGLE THEOREM.
We observe now that because of Proposition (iv), the original
propositions about circles can be rephrased as a single, somewhat mystery
existence theorem about points lying within a triangle.
TRIANGLE THEOREM. Let a triangle have vertices at points
A, B and C, as in Figure 9.
(i). There exists a unique interior point D, for which
the three line segments emanating from the vertices and passing through
D, intersect 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 and
positive number r, for which three line segments emanating from the
vertices to F have lengths, WHEN SHORTENED BY r, given by
Ab, Bc and Ca.
(iv).
The interior points D, E and F are coincident
only for equilateral triangles.
Figure 9:
2329.
PROOF (SKETCH).
Part (i) follows from a very old theorem of Ceva (1678),
where (using area arguments) it requires that the products of alternating
lengths, (Ab)(Ca)(Bc) and (aB)(cA)(bC) be equal. In Chapter 8,
we sketch a direct, analysis-based proof for this special case, motivated
by the argument employed above to establish Proposition (iv).
Part (ii) follows from Part (i) and Figure 10,
where E necessarily emerges as a shared vertex of two pairs
of congruent right triangles.
Part (iii), substantiated (arithmetically) in other sections,
is illustrated in Figure 11.
We shall explore this difficult construction in
Chapter 6.
Part (iv) is clear observationally, as seen in Figures 1
and 2. (The reader may wish to add complete arguments.)
Figure 10:
2330.
Figure 11:
2331.
CHAPTER 4.
INTERNAL TANGENCIES.
The Triangle Theorem, reflecting as it does the consequences of mutually
tangent circles centered at the vertices of triangle ABC having only
external tangencies, does not carry over to the alternative configurations
resulting from internal tangencies, and where the meeting points lie
outside of the triangle. However, analogous arguments establishing
the existence of two of these special points do carry over, as demonstrated
in Figure 12.
Figure 12:
2332.
In fact, the meeting point E again necessarily emerges as the shared
vertex of two pairs of congruent right triangles. Also, the meeting point
F emerges as the center of a circle of radius r, the value of
which is determined from the fact that radial segments of length r
are portions of the edges of triangles sharing the common edges CF
and AB. So the area of the quadrangle ACBF becomes that of the
combined areas of two pairs of two subtriangles. The meeting point D
(displayed in a different circumstance here for clarity), lying outside
the triangle ABaC, conforms to a new version of that
concurrency problem, whose formulation is examined in Chapter 8. One should
also note that in the alternative configuration case, with each triangle
ABC there are three separate configurations to contend with, depending
upon which vertex becomes the center of the large circle enclosing the other
two. Consequently, there are also three sets of meeting-points
to contend with.
CHAPTER 5.
DESCENDING CASCADES OF TRIANGLES.
A fascinating aspect of the geometry is borne out pictorially upon
successively applying Proposition (iv), in turn, to the
resulting triangles (such as abc) which are determined by the
tangency-points. These triangles cascade downward upon some interior point
P, as illustrated in Figure 13.
Figure 13:
2333.
In the particular case illustrated in Figure 13, the triangle
ABC is the absolute "top" of the cascade. For points A,
B, and C can NOT be the tangency-points of a larger
triangle displaying A, B, and C along its edges, unlike
those further down the cascade. Also, we observe that the special interior
point D of triangle ABC (fulfilling Part (i))
is distinct from P.
It is clear that equilateral triangles do not possess natural "tops", and so,
logically their cascades begin at infinity. This strange, singular phenomenon
is easily explained by further observations, that any acute triangle is not
(cannot be) the "top" of a cascade (there is always a larger one), while any
obtuse or right triangle is necessarily (must be) a "top" (as there can be
no larger one). In fact, there exists a larger one, only if its vertices
lie on the perpendicular bisectors of the edges of its antecedent.
Figure 14:
2334.
As illustrated in Figure 14, a triangle ABC, with vertices on
the perpendicular bisectors of its acute antecedent abc can always be
constructed. This construction is possible because the parallels to
the upper perpendicular bisector passing through vertices b and
c, intersect the lower ones at points e and f, so that
segment ef passes above the largest-angled vertex a.
Thus, a line parallel to ef and passing through vertex a,
determines approximate locations of vertices C and B,
as intersections with these lower perpendiculars. These, in turn, determine
an approximate location for vertex A, some distance above, where
b and c lie on the edges CA and BA. Then one
observes that rotations of this latter parallel line (through a) moves
the approximate location of A from one upper parallel to the other
(as determined by points e and f). So the exact larger triangle
is determined by the particular rotation which places vertex A on the
upper prependicular. The altitude (in this case) from the vertex A
is approximately 4.7 times those from vertices B and C,
guaranteeing that the next step-up triangle is obtuse.
Finally we observe that if segment ef lies on or below vertex
a, such as when angle a is right-angled or obtuse, then the
construction of vertex A certainly cannot be completed.
Triangle abc is, then, a top one for the cascade.
The particular illustration in Figure 14 has been specially selected
to demonstrate that the D-points of successive acute triangles in a
cascade need not be coincident, although quite generally, graphic
distinctions cannot be detected. (Note that even the smaller acute triangles
in this cascade appear to share the same point d.) So only one obtuse
or right triangle (the "top") can belong to a cascade, and its special
interior point D does not coincide with the limiting interior point
P of the remainder of the cascade, necessarily consisting
of descending acute triangles.
Another interesting observation obtained from the pictures is that the
triangles in all these cascades rapidly tend to equilateral triangles
(toward the bottom), so that the ratios of the areas of that of the
triple-spiked regions to the areas of the triangles must converge to the
precise value [1 - π/2 √3], noted above.
CHAPTER 6.
OTHER VIEWS OF
DESCENDING CASCADES.
I. Since Proposition (iv) concerns mutually tangent circles,
one might imagine a replacement of the triangles in the descending cascade in
Figure 13, by corresponding,
mutually tangent
circles. This replacement produces a very
attractive, microscopic display of cloverleafs of mutually tangent circles
(using a right-angled top-triangle, for enhancement purposes), cascading
downward to the point P.
Figure 15:
2335.
II. Another view of these interesting cascades is obtained by employing
the perpendiculars from the E-points of Part (ii).
Figure 16 illustrates the appearance of such a descending cascade,
demonstrating even more clearly, that the triangles converge down upon the
interior point P.
Figure 16:
2336.
III. Finally, we consider the appearance of these cascades obtained by
employing, successively, the embedded circles of Part (iii) (also
using a right-angled top-triangle). As illustrated in Figure 17,
Figure 17:
2337.
the circles and the centers (F-points) converge down upon the interior
point P. Here, the mutually tangent arcs are clearly on display,
as well as the embedded circles.
IV. The step-up from such acute triangles can always be made, just as
in the initial cascades. However, locating a step-up F-point requires
a further construction, using mutually tangent arcs, as in Figure 11.
An embedded circle, of course, is to be chosen tangent to the arcs.
Previously this has been mainly an "imagined" construction. So we shall
outline an iterational (approximation) scheme, which likely converges
to F.
Figure 18:
2338.
From the mid-point between points D and E (i.e., the first
estimate for F), perpendiculars to the arcs locate potential
tangency-points. Then rays from the vertices through these latter points
may not be concurrent, and if not, an improved estimate for F can
be chosen as some central point within the small, induced triangle (greatly
exaggerated in Figure 18), in order to equalize the small radii.
If they are concurrent, of course, then one has obtained F.
The starting estimate in this scheme could be any other reasonable choice,
but it is only with extremely obtuse triangles that F deviates very
far from the mid-point between D and E. In many respects,
F should be considered the truly natural center for a triangle,
and begs for some sort of direct geometrical construction.
SOME REFLECTIONS.
V. These three views of the families of cascades, with triangles
approximating equilateral ones downward and converging down upon undetermined
limits, and with obtuse or right-angled triangular "tops", quite naturally
lead to the separation of all acute triangles according to whether their
step-up triangles are acute or not. The isosceles ones readily make a
distinction, according to the value R of the ratio of the largest edge
to the shortest edge. In order for an isosceles triangle to possess an acute
step-up one, its R-value must not exceed the bound
(1/2)[1/(√2 - 1)2 + 1]1/2 ~ 1.3065.
This bound is simply the R-value of an isosceles triangle, whose step-up
triangle is right-angled. It can be derived directly from the relevant
geometric configuration. (Readers are encouraged to derive this formula.)
Assuming that all acute triangles can be similarly distinguished,
Figure 19 illustrates that for the general situation, the limiting
value (separating the acute triangles) might be only slightly larger than
that given by the above bound. For here is an obtuse step-up triangle
stemming from an acute antecedent with R measuring approximately 1.44.
On the other hand, for the specially selected example in Figure 14,
the ratio R for the acute triangle abc also measures
approximately 1.44, but where the step-up triangle ABC remains
acute. This seems to establish a rather likely (approximate) separation value
of 1.44.
Figure 19:
2339.
Whatever the value, it becomes something of a measure of the necessary
symmetry (R=1 for equilateral triangles) required of an acute triangle
in order for it to continue to participate in step-up constructions.
We now have a clear explanation for the observed uniformity of the descending
triangles, where after just a step or two, all appear to be essentially
equilateral ones, and seem to share the same D-point. This rather
interesting situation is explored in [5].
VI. Incidentally, such considerations do not apply to cascades
of medial triangles, where the obtuse and acute collections never overlap one
another. In the traditional literature, the D-points of
Part (i) go by the name of Gergonne ([3], page 13) and are
lumped in with the many special points of synthetic geometry. Although given
a prominent position, their intimate connections with mutually tangent
circles seems to be largely ignored, as well as the circles themselves,
and even more so, the triple-spiked regions. Knowledgeable readers may note
the lack of interest here in incircles, circumcircles and excircles,
which are rendered pivotal positions in the literature. In our treatment,
these just become distractions, as we are definitely interested in circles,
but not in these particular circles.
VII. It seems certain that F is never the actual mid-point
of the segment DE, but a rigorous proof of this assertion remains
elusive, as well as that of the surmised collinearity of D, E
and F. It appears that the line DE may be parallel to the
Euler-line [3,4].
CHAPTER 7.
A THREE-DIMENSIONAL VIEW.
I. An interesting (and very rewarding) three-dimensional view concerns
three non-intersecting SPHERES which are tangent (externally)
to one another. This results in more general, comprehensive observations,
(i) the three line-segments drawn from the centers to opposing
tangency-points meet at a point,
(ii) the three tangency-planes meet along a line
and at a point lying on the centers' plane, and
(iii) these two meeting points are coincident only if the
three mutually tangent spheres possess equal radii. Only then are they
coincident with the center of the largest sphere that can be embedded
within the (internal) region determined by the mutually tangent spheres.
This three-dimensional view is best imagined as being focused at the center
F of the embedded sphere. The three mutually tangent spheres can
then be pictured as "rolling around" the embedded sphere, creating other
(concentric) spheres traced out by the centers, the tangency-points and the
meeting-points. In the most general situation, there are nine distinguishable
concentric spheres created, including the embedded one. In the completely
symmetrical situation, there are only three. In this expanded view,
the embedded sphere clearly becomes central to the entire picture.
Consequently, it becomes appropriate first to designate a particular sphere
as the central "embedded" one, and only then to consider various collections
of four, mutually tangent spheres, where the outer three must
be chosen larger than the designated one. One of these spheres can be chosen
arbitrarily, but leads to definite restrictions on choices of the other two.
The spheres can then roll around the designated central sphere, creating
the concentric spheres that we envision.
Such a view carries with it the corresponding observations concerning sprays
of space triangles, with their direct associations reflected in the center's
concentric spheres. Proposition (iv), of course, requires only
a change of word, "circle" to "sphere", while all the Figures 1-8, 15,
and 17, similarly become re-interpreted as merely showing traces,
on planes, of spheres and planes, for a single generating configuration
of three mutually tangent spheres.
II. Continuing our general observations, we note that, in the case of
an acute centers' triangle, of the nine spheres discussed above, the outer
three are center-point spheres, the next three are tangent-point spheres,
followed by the (central) embedded sphere, and finally the two meeting-point
spheres. With isosceles center triangles, some of these coalesce, and with
equilateral center triangles, they reduce down to just three. In the case of
an obtuse centers' triangle, the meeting-point spheres may change places with
the central sphere (see Figure 8). One requires some very artistic
displays of these phenomena, unavailable to the writer, for a proper
perspective. So we show none.
Descending cascading takes place simply by dropping the outer three
(center-point) spheres, replacing them with the three tangent-point spheres,
forming new tangent-point spheres (using Proposition (iv)), and
finally constructing the three (resulting) smaller ones (following the
Triangle Theorem). These descending cascading spheres ultimately converge
down onto the elusive limit point P.
III. In consequences I and II of Chapter 1, we have
imagined animated-type programs showing the evolution of figures, as the
sizes of the circles change. So we suggest here, that similarly imagined
three-dimensional versions of the nine spheres processes with changing radii,
would also be quite intriguing. This is particularly true for the
generation and the movements of the concentric spheres up and down a cascade,
with total collapse down upon P (Recall the traces of generating
configurations in Figures 15 and 17). What an exciting,
colorful display could be produced for television!
CHAPTER 8.
ADDITIONAL CONCURRENCY CONSIDERATIONS.
I. Our direct proof of Part (i) of the Triangle Theorem
stems from simple observations concerning the evident geometry exhibited by
continuous families of triangles and circles. In Figure 20,
Figure 20:
2340.
we observe a triangle ABC with a circular arc centered at its
largest angle A which is too large to allow for mutual tangency
with those centered at B and C. The resulting crossing point
D, of rays from the latter vertices, appears to fall well within
triangle Aad. That this is necessarily the case, follows from the
observation that the given triangle ABC of Figure 20
is intermediate between an isosceles triangle and a "virtual" triangle
(when B →∞), as shown in Figure 21.
Figure 21:
2341.
For the (dashed) former, D lies on the mid-line from A
(because of symmetry), and D then remains within the Aad
triangles for all finite B, reaching the fixed edge Ad only for
B = ∞. Therefore, the crossing-point D certainly always
falls within triangle Aad, and becomes the concurrent one of
Part (i), with the "correct" circle at A as the edges
Aa and Ad coalesce.
II. Interestingly enough, the limiting circumstance of the above proof
leads to a new concurrence problem for a rectangle (not amenable
to Ceva's Theorem), illustrated in Figure 22.
Figure 22:
2342.
Readers may wish to attack this elusive problem, where the lengths
of Ab and Ac, and of Cb and Cd, are equal.
For our part, we make some simple observations suggesting the result.
As illustrated in Figure 23,
Figure 23:
2343.
Aℓ is a point where the length of the upper segment
Aℓbℓ clearly exceeds that of the
lower segment Aℓc, while the length of
Cdℓ equals that of Cbℓ,
and the point Au, lying above Aℓ,
completes the concurrency condition. We then observe that the length
of the lower segment Auc now exceeds that of the
upper segment Aubu.
Thus as Aℓ advances upward, Au
descends, and all requirements of Figure 22 are fulfilled when they
meet. In this construction, they DO meet, and actually change places
as Aℓ continues to advance upward.
III. The new version of the concurrency problem with internal
tangencies (Figure 12), and which produces a meeting point
D
outside triangle
ABC,
has been reformulated in Figure 24 for a required isosceles triangle
(ultimately, for a quadrangle).
Figure 24:
2344.
With circle centers at A and B leading to an external tangency
at H and internal tangencies from the large circle at C
at the base angles, the objective is to prove that the double crossing-point
D partitions the segment passing through D and parallel
to AB, in the ratio x to y. For then the ray from
C to D will pass through H, as demanded of the
triple crossing-point. Readers may wish to attack this elusive concurrency
problem. For our part, we make the simple observation that the problem
concerns only the quadrangle as labeled. Center C in the picture can
be replaced by the two equal angles at the base. This is then specific
enough to suggest a resolution for any positive x, y and
z, satisfying 2(xy)1/2 < z < 2(x+y).
(A proof appears to be extremely challenging.)
CHAPTER 9.
ASCENDING CASCADES.
We conclude with some intriguing observations concerning the resulting
(alternative) cascades based upon the mutually tangent circle configurations
as illustrated in Figure 3. By employing the very same parallel line
arguments that were used in connection with Figure 14,
this alternative process must result in unending ascending cascades
possessing limiting "bottoms", consisting of acute or right-angle
triangles, with all ascending members being obtuse triangles. This is
certainly something of a pleasing counterpart to the descending cascades
of Figure 13, with their obtuse or right-angled "tops" and all
descending members being acute triangles. The relevant construction
(replacing tangency points with circle centers) is a clearly possible
and valid procedure (the counterpart to Proposition (iv)).
These ascending cascades, however, exhibit much greater variety
due to the fact that the choice of tangency points to be
replaced by the centers of the largest circles (enclosing the other two)
is optional. This also leads to much more interesting movements of the
meeting points D, E and F roundabout the ascending
obtuse triangles. In a descending cascade, everything simply collapses
down upon the limiting point P.
One can now imagine linking together an ascending and a descending cascade
of mutually tangent circles into a single unit climbing from P
to ∞. Such a linked cascade is illustrated in Figure 25
Figure 25:
2345.
showing the change from Figure 15, at a chosen centers' triangle,
ABC, by replacing the (dashed) externally tangent circles with the
first set of mixed tangency circles advancing upward. As indicated,
the transition can be made at any point along the way (in either portion),
and is not limited to simply connecting a "top" with a "bottom", although
that circumstance is certainly possible. One can certainly imagine much more
elaborate and numerous interchanges of ascending and descending circles
and triangles. These might be models of biological growth and decay
processes, such as in the skin, the bone marrow, or in the digestive tract.
Other applications might be envisioned in cell growth phenomena
[7,8].
Our final observation, reflected in Figure 25, leads to a
clarification of the mechanism of the linking process; it is simply that
of an interchange of the circles centered at A and B, followed
by an accommodating adjustment of the size of the circle centered at
C. This clarification is also reflected
in Figure 24 by an
analogous interchange of the internal Gergonne point G with the
external concurrency point D for the triangle ABC.
The mechanism is again an interchange of the two radii x and y
at A and B followed by an adjustment of the radius at C.
It is of interest to note that this radius of the largest circle is equal
to the sum (x+y+w) of the radii of the three smaller circles.
(Their sum is, in fact, Euler's familiar half-perimeter, s.)
Should the other two vertices be used as the centers for the largest circles,
resulting in two other external concurrent D-points, then the
resulting rays CD, BD and AD become concurrent
by Ceva's theorem. The reader is encouraged to establish this fact, upon
constructing the other two configurations, suggested in Figure 24.
ACKNOWLEDGEMENTS.
I wish to thank G. William Moore, MD, PhD, for reviewing and formatting
the manuscript, and G. Vincent Moore for providing the graphic art. I further
wish to thank my wife of sixty-some years, Marilyn Struble, who rekindled
in me a dormant enthusiasm in mathematics, after a sixteen-year hiatus.
She had the insight to give me stimulating books on this and related topics
in mathematics.
REFERENCES.
1. Struble RA.
Elementary problems about circles.
Infinite products, filling programs, and integration.
Preparation completed.
2. Chandler RE, Meyer CE, Rose NJ.
Eudoxus meets Cayley.
Amer Math Monthly. 2003;110:912-927.
3. Coxeter HSM, Greitzer SL.
Geometry Revisited. New Mathematical Library.
Washington, DC: Math Assoc Amer. 1967.
ISBN: 0883856190, 207 pages.
4. Honsberger R.
Episodes in Nineteenth and Twentieth Century Euclidean Geometry.
New Mathematical Library. Washington DC: Math Assn America. 1996.
Second printing. 2005.
ISBN: 0883856395, 174 pages.
5. Struble RA.
Cascades of Triangles Sharing Lines through Vertices.
Preparation completed.
6. Struble RA.
A New Direction in Geometry.
Submitted.
7. Moore GW, Struble RA, Brown LA, Kao GF, Hutchins GM.
Cell Surface Tessellation: Model for Malignant Growth. (Abstract).
Arch Pathol Lab Med. 2007 Jun;131:.
http://apiii.upmc.edu/abstracts/posterarchive/2006/eposter/moore.html
http://www.netautopsy.org/celltess.htm
8. Moore GW, Struble RA, Brown LA, Kao GF, Hutchins GM.
Triple-spiked Zones in Cell Surface Tessellations:
Model for Malignant Growth. (Abstract).
Arch Pathol Lab Med. 2008 Jun;132:. in press.
http://apiii.upmc.edu/abstracts/display_07.cfm?id=376
http://www.netautopsy.org/triplspk.htm
CONTACT INFORMATION.
Raimond A. Struble
Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh
Raleigh, NC.
email: raimondstruble@yahoo.com
Last updated: 8/30/2008, by Raimond A. Struble, PhD.