Synthetic Geometry Stemming From A Central Circle. 10/31/2006.

SYNTHETIC GEOMETRY
STEMMING FROM A CENTRAL CIRCLE.
DRAFT COPY ONLY.
10/31/2006.
Raimond A. Struble.

Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh
Raleigh, NC.
http://www.infiniteproduct.info/strusynt.htm

Send comments and correspondence to: raimondstruble@yahoo.com

Mathematics Review Subject Classification Number: 51M04.

ABSTRACT.

INTRODUCTION.

The work in [3] suggests that the logical center of a triangle is that of the circle embedded within the three mutually tangent circles centered at the vertices of the triangle. It is our purpose here to develop this point-of-view by revealing the geometrical mechanisms by which a given circle actually becomes an embedded one. Perhaps this is a novel point-of-view, overdue and of some interest at least pictorially. The development is mainly synthetic, but supported with a considerable amount of analytic geometry.

I. THE CENTER'S LINE.

We begin with a touch of analytic geometry, by selecting a unit circle centered at the origin, and a circle of radius a, (greater than 1), centered along the x-axis, and tangent to the unit circle. We are interested in the other circles tangent to these two, and especially the locations of their centers.

FIGURE 1.

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It is clear that the centers lie along a curve (the centers' curve), stretching from the x-axis, to infinity.

FIGURE 2.

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The center's curve extends above and below the x-axis, reminiscent of a hyperbola, and lies within channels determined by lines which are perpendicular to the common tangents to the first two circles.

FIGURE 3.

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The center of any third (tangent) circle is located along two radii stemming from the centers of the first two circles, which also locate the tangency points.

FIGURE 4.

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A recent investigation [5] shows that the tangency points are collinear with the intersection, T, of the common tangents and the x-axis. However, the converse also holds, so the complete centers' curve can be constructed synthetically, using rays from T passing through the first two circles, and noting the interior points of intersection.

II. THE GERGONNE LINE.

The work in [4] shows that if the same construction is applied to pairs of all the tangency points, then the three meeting points are collinear.

FIGURE 5.

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This circumstance represents the result as it applies to the (concurrent) Gergonne point [1], so we shall label the meeting line as the Gergonne line.

III. SOME ANALYTIC GEOMETRY.

It is instructive to venture further into some analytic geometry, employing the usual Cartesian coordinates, (x,y) along the centers' curve.

FIGURE 6.

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The area of the relevant triangle is clearly given by

A = 1/2 y (1+a).

However, by using Young's formula [2], the square of the area is given by

A² = ab(1+a+b).

These two formulas imply that the third radius is necessarily given by

b = (1+a)/2 [√(1+y²/a) - 1]

Therefore, the Cartesian equation of the centers' curve becomes

x² + y² = (1+b)² = {1 + (1+a)/2 [√(1+y²/a) - 1]}²

A more revealing equation results from the elimination of y, which leads to

x² = (1-2a) b² / (1+a)² - 2a² b / (1+a)² + 1.

In the (x,b) coordinates, this is the equation of a hyperbola. Indeed, with the scale change, β = [√(2a-1)/(1+a)]b, the last equation becomes

[β+a/(1+a)√(2a-1)]² - [x - 1/2]² = a⁴/[ (1+a)²(2a-1)] - 1/4,

where the right-hand member varies from 0 to ∞ as radius a varies from 1 to ∞. But hyperbolas possess two branches, so what transpires with the other one? This is just another centers' line which is reflected in the y-axis, when the a-circle lies to the left of the unit circle. In the (x,y) coordinates, these two branches cross, of course, as they surround the unit circle.

The importance of this analytical formulation lies in the difficulty one has in using the synthetic formulation for large, third circles. The tangency points tend to become obscure, and arithmetic can come to the rescue. This is significant, since it is only with the larger values of the third radius b that the unit circle becomes a potential embedded circle.

III. THE EMBEDDING OF THE UNIT CIRCLE.

There is a special value b_a for which the third circle becomes tangent to one of the common tangent lines.

FIGURE 7.

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For values of b less than b_a, the unit circle cannot be an embedded circle lying within three mutually tangent circles. (It can only be an external one of four mutually tangent circles.) For this critical value, the relevant common tangent line becomes the corresponding Gergonne line for this triangle, and the third circle is then tangent to it. Only for larger b-values will the third circle penetrate below the common tangent line so as to be available for a fourth mutually tangent circle coming from the lower portion of the centers' curve.

The crossing-point T of Figure 7 is located equidistant from the tangency point t and the tangency point between the first two circles [5]. This common distance d = 2a/(a-1) can be obtained using two versions of "sin θ" on the right. It turns out that the center C of the b_a-circle lies not far removed from the angle bisector of the two common tangency lines (the dashed line). Now

(TC)² = b_a² + d²,

while

(TC_e)² = ( b_a + a + (CC_e)² - (d+a)²

If C_e were C, then this would give

b_a ~ d(d+a)/a = 2a (a+1) / (a-1)²,

a slightly large value. However, this estimate is quite useful, since any larger b-value clearly heralds the possible embedding of the unit circle, and the exact value of b_a can only be obtained by solving some horrendous simultaneous equations. On the other hand, Figure 7 supplies a synthetic approximate location of the center of the b_a-circle. As the b-circle continues to increase in size, the fourth circle (producing the embedding) peels away from the lower common tangent line (representing a circle of infinite radius), and requires an extremely large radius.

FIGURE 8.

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Here arithmetic coming from analytic geometry becomes useful, even necessary, in determining a fourth radius. With the appearance of this fourth circle, the original centers' triangle (circles one, two, and three) suddenly gives away to the embedding centers' triangle (circles two, three, and four). An interesting occurrence in itself, as though the fourth circle springs into being from its center on the lower centers' curve at infinity. With continued increases in the b-radius it ultimately attains a LAST value b_L, as the b-circle becomes tangent to the x-axis, and the fourth circle touches and matches the third circle.

FIGURE 9.

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The last center's triangle becomes isosceles, morphing from a very unsymmetrical triangle, which may be either obtuse (Figure 10) or acute (Figure 9). The last value, b_L, of the second radius can be obtained from the two associated right-triangles by eliminating L. Eliminating L from the resulting two equations

b_L = (L²-1)/2
EQ: b_L = [(L+a+1)²-a²]/2a

is certainly possible, but somwhat impractical, and furnishes little useful indication as to the influence on b_L of the size of radius a. A useful approximation is clearly in order, and can be obtained by noting that the crossing point T is not very far removed from the mid-point of the segment of length L. Since d = 2a / (a-1), this approximation results in a simple estimate,

L ~ 2 ((a+1)/(a-1)),

which can be used in the above two formulas to obtain the approximate b_L-values

b_L ~ [ 4 ((a+1)/(a-1))² - 1 ] /2
and
b_L ~ {[ 2 ((a+1)/(a-1)) + a + 1 ]² - a²} / 2a.

The reader may with to derive an explicit average formula for b_L from these. Fortunately, one of these estimates will be slightly too large, and the other will be slightly too small, depending upon whether the approximate L-value is too large or too small. Therefore, the average of the two approximations can be expected to yield reasonably good estimates for the final radius. Although of some theoretical interest to this development, the exact final radius is merely the one producing an isosceles triangle leading to the embedding of the unit circle. This final radius, therefore, is obtainable by a simple scale change for any known example. The reader is invited to uncover prescriptions for these examples somewhere, and perhaps settle the issue once and for all. Improved arithmetical values are readily obtained by adjusting the numerical value of the number L until the two equations in EQ yield the same numerical value of b_L.

IV. THE TRIANGLES.

ALL the accompanying embedding triangles encountered here share the same central (unit) circle, as they modulate according to the b-values between b_a and b_L, which themselves depend upon the starting a-value, of course.

The possible last centers' triangles so obtained, also depend upon the starting a-value. In order to obtain an equilateral triangle, for example, the radius a must be exactly equal to √3 / (2 - √3) [6]. As the size of the third circle increases, its radius b passes the critical value (embedding the unit circle) and eventually reaches the same value, yielding the equilateral triangle. With larger initial a-values, one always obtains a final acute isosceles triangle; and with sufficiently smaller initial a-values, one obtains final obtuse triangles (Figure 11).

FIGURE 10.

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V. OLD AND NEW SYNTHETIC GEOMETRY.

At this point, one can invoke all the standard synthetic geometry relating to triangles that is currently available, with the addition of one new challenging (central) point, the origin. This point is the logical CENTER of any triangle, and so shall be labeled.

The unit center circle is always embedded tangentially within three unique, mutually tangent circles, centered at the vertices of the triangle. The radii a, b and c of the embedding circles necessarily satisfy the inequalities a>1, b_a < b < b_L (where b_a and b_L depend upon a), and c > b just accommodates the embedding process.

We propose one new theorem for synthetic geometry (suggested by [3]):

THEOREM 1. For any triangle, the line-segment between the Gergonne point and the incenter passes through the center of the triangle, and is parallel to the Euler line.

Other new theorems concerning the center and the center circle seem to demand articulation and proofs. Such theorems could conceivably relate to continuous morphings of triangles about a (fixed) center circle and the many special points, lines, and circles of current theory. These will move in-and-out about in fascinating patterns. For example, the motion of the Gergonne line following triangle variations will trace out intriguing curves as tangent lines, such as witnessed in [4]. Synthetic geometry need not be restricted to static configurations when these configurations relate to one another in a one-to-one fashion. So why not study the morphing of familiar configurations about a (fixed) center circle? Surely much can be learned, and concurrency results involving the center are more likely to be discovered this way than in the vacuum presently created by what appears to be total neglect of the center circle.

REFERENCES.

1. Coxeter HSM, Greitzer SL.
Geometry Revisited.
New Mathematical Library.
Washington, DC: Math Assoc Amer. 1967;:4-5.
ISBN: 0883856190, 207 pages.

2. 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.

3. Struble RA.
Observations about Mutually Tangent Circles and Related Triangles.
In preparation.

4. Struble RA.
A new direction in Euclidean geometry.

6. Struble RA.
An Elementary Problem about Circles. Infinite Products, Filling Programs, and Integration.
Preparation completed.

CONTACT INFORMATION.

Raimond A. Struble
Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh
Raleigh, NC.

1431 Ocean Avenue Apt #410
Santa Monica CA 90401

Voice: 310-995-1768
Fax: 310-458-1768

email: raimondstruble@yahoo.com

Last updated: 10/31/2006, by Raimond A. Struble, PhD.

SYNTHETIC GEOMETRY STEMMING FROM A CENTRAL CIRCLE. DRAFT COPY ONLY. 10/31/2006. Raimond A. Struble.

Professor Emeritus Department of Mathematics North Carolina State University at Raleigh Raleigh, NC. http://www.infiniteproduct.info/strusynt.htm