Apollonius’ Problem

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David Gisch and Jason M. Ribando Department of Mathematics University Northern Iowa Cedar Falls, Iowa 50614-0506 USA
Figure 7a.
Figure 9.
Figure 10b

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Apollonius’ Problem: A Study of Solutions and

Their Connections

David Gisch and Jason M. Ribando

Department of Mathematics

University Northern Iowa

Cedar Falls, Iowa 50614-0506 USA

Received: August 15, 2003

Accepted: February 29, 2004

ABSTRACT

In Tangencies Apollonius of Perga showed how to construct a circle that is tangent to three given

circles. More generally, Apollonius' problem asks to construct the circle which is tangent to any

three objects that may be any combination of points, lines, and circles. The case when all three

objects are circles is the most complicated case since up to eight solution circles are possible

depending on the arrangement of the given circles. Within the last two centuries, solutions have

been given by J. D. Gergonne in 1816, by Frederick Soddy in 1936, and most recently by David

Eppstein in 2001. In this report, we illustrate the solution using the geometry software

Cinderella™, survey some connections among the three solutions, and provide a framework for

further study.

I. INTRODUCTION

Apollonius of Perga was known as

'The Great Geometer'. He should not be

confused with other Greek scholars named

Apollonius, for it was a common name.

Little is known of his life except that he was

born in Perga, Pamphylia, which today is

known as Murtina, or Murtana, and is now in

Antalya, Turkey. The years 262 to 190 B.C.

have been suggested for his life [1-3]. It is

commonly believed that Apollonius went to

Alexandria where he studied under the

followers of Euclid and possibly taught there

later.

This paper focuses on a problem

solved by Apollonius in his book

Tangencies. Apollonius’ works have had a

great influence on the development of

mathematics [4]. In particular, his famous

book Conics introduced terms which are

familiar to us today such as parabola, ellipse

and hyperbola. In Book IV of the Elements,

Euclid details how to construct a circle

tangent to three sides of a given triangle

(Proposition 4) and how to construct a circle

containing three noncollinear points

(Proposition 5) [5, p. 182]. The latter

construction is accomplished by finding the

intersection point of the perpendicular

bisectors of any two sides of the triangle

with the three given points as vertices. In

Tangencies Apollonius poses a

generalization to Euclid’s two propositions:

given any three points, lines or circles in the

plane, construct a circle which contains the

points and is tangent to the lines and circles.

Apollonius enumerated the ten combinations

of points, lines and circles and solved the

cases not already solved by Euclid [6,

p.182]. The case when all three objects are

circles is the most complicated of the ten

cases since up to eight solution circles are

possible depending on the arrangement of

the circles (see Figure 1).

Since no copy of Apollonius’

Tangencies has survived the ages, Pappus

of Alexandria deserves credit for eternally

linking Apollonius’ name with the tangents

problem. Pappus, who lived some five

centuries after Apollonius and is known

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Figure 1. Eight solution circles to Apollonius’ problem.

mostly for his encyclopedic recording and

commenting of Greek mathematics, wrote a

“Treasury of Analysis” in Book VII of his

Mathematical Collections dedicated to works

of Euclid, Aristaeus, and Apollonius. Here

he succinctly states Apollonius’ problem,

acknowledges the ten cases, and provides a

compass and straightedge solution for at

least one solution circle [6, p. 182].

Excepting Arabic reconstructions of

Apollonius’ works, Apollonius’ problem lay

dormant in the literature until François Viète

(1540-1603) restored the Tangencies in

1603. Having unveiled Apollonius’ solution,

Viète challenged Adrianus Romanus to draw

a circle tangent to three given circles but

was disappointed by Romanus’ use of

conics. A century later Newton also went

beyond compass and straightedge solutions

by employing hyperbolas [7]. Agreeing with

Viète’s preference for compass and

straightedge solutions, we are motivated to

include three related constructions under the

same cover: the Euler-Gergonne-Soddy

triangle, which contains the centers of the

two solution circles in the special case when

the circles are mutually tangent or “kissing;”

a solution found by David Eppstein in 2001

for the same special case; and an

adaptation to Gergonne’s analytic solution

which constructs all eight solution circles in

the general case for three circles in generic

position. We conclude with some

connections among the three solutions and

provide a framework for further study

beyond the scope of this paper.

II. THE

EULER-GERGONNE-SODDY

TRIANGLE OF A TRIANGLE

A special case of Apollonius'

problem is known today as the three coins

problem, or kissing coins problem. In this

variant, the three circles, of possibly different

radii, are taken to be mutually tangent.

There are two solutions to this special case

of Apollonius’ problem: a small circle where

all three given circles are externally tangent,

and a large circle where the three given

circles are internally tangent. In 1643 Renè

Descartes sent a letter to Princess Elisabeth

of Bohemia in which he provided a solution

to this special case of Apollonius’ problem.

His solution became known as Descartes’

circle theorem. Philip Beecroft, an English

amateur mathematician, rediscovered

Descartes’ circle theorem in 1842. Then it

was discovered again in 1936 by Frederick

Soddy (1877-1956), who had won a Nobel

Prize in 1921 for his discovery of isotopes

[8]. Soddy expressed the theorem in the

form of a poem, "The Kiss Precise," which

was published in the journal Nature and is

included below. It may have been the flavor

of the added poem that set Soddy apart

from his predecessors, as the two circles are

known today as the inner and outer Soddy

circles. Additionally, Soddy extended the

theorem to the analogous formula for six

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

spheres in three dimensions [9, Soddy

Circles].

Though Soddy provided an analytic

solution to the kissing coins problem, we do

not know whether he constructed a synthetic

solution using compass and straightedge.

For our part we find compass and

straightedge solutions easier to understand

than their analytic counterparts. This

section is devoted to the Euler-Gergonne-

Soddy triangle (EGST). Though it does not

provide a solution to the kissing coins

problem, it does provide a unique insight to

finding the Soddy circles. Many of the

connections between the EGST and the

Soddy circles solution to the kissing coins

problem were inspired by Oldknow’s article

[8], which uses trilinear coordinates,

harmonic ranges, and parameterizations of

lines to construct the EGST. It is interesting

to note that Oldknow attributes many of his

investigations to the use of geometric

software packages, a current trend among

geometry researchers.

As noted already, the kissing coins

problem is the special case of Apollonius’

problem where all three circles are mutually

tangent. In this arrangement there are two

solution circles, the inner and outer Soddy

circles as shown in Figure 2. There are two

questions of interest that arise from this

special case: what are the radii of the Soddy

circles and where are their centers located?

As a partial answer to the latter

question, the Soddy centers lie on the line

Figure 2. Soddy circles.

The Kiss Precise

For pairs of lips to kiss maybe

Involves no trigonometry.

'Tis not so when four circles kiss

Each one the other three.

To bring this off the four must be

As three in one or one in three.

If one in three, beyond a doubt

Each gets three kisses from without.

If three in one, then is that one

Thrice kissed internally.

Four circles to the kissing come.

The smaller are the benter.

The bend is just the inverse of

The distance from the center.

Though their intrigue left Euclid dumb,

There's now no need for rule of thumb.

Since zero bend's a dead straight line

And concave bends have minus sign,

*The sum of the squares of all four bends

Is half the square of their sum.*

To spy out spherical affairs

An oscular surveyor

Might find the task laborious,

The sphere is much the gayer,

And now besides the pair of pairs

A fifth sphere in the kissing shares.

Yet, signs and zero as before,

For each to kiss the other four

*The square of the sum of all five bends

Is thrice the sum of their squares.*

Frederick Soddy (Nobel Prize, Chemistry,

1921) Nature, June 20, 1936 [10].

determined by the incenter and Gergonne

point of the reference triangle. This line is

known as the Soddy line and it contains one

of the sides of the EGST. Given three

vertices of a triangle, we show how to

construct its EGST. The reader may want to

refer to the Appendix for a glossary of

geometric terms involved in this

construction.

We start with the Euler line since it

is the easiest to construct. Given a

reference triangle, find the circumcenter as

the intersection of the perpendicular

bisectors and the orthocenter as the

intersection of the altitudes. These two

points determine the Euler line, which also

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Figure 3. The Gergonne point, incenter, incircle and Soddy line.

contains the triangle’s centroid (the

intersection of the medians). The Euler line

also contains a number of other important

triangle centers including the center of the

nine-point circle.

The Soddy line is determined by the

incenter, where the angle bisectors coincide,

and the Gergonne point. The incenter is the

center of the unique circle that is internally

tangent to its reference triangle at three

points. These three points are called the

contact points, and together they form the

contact triangle of the reference triangle.

The Gergonne point can most easily be

found as follows: erect perpendiculars

containing the incenter from each side of the

reference triangle. These perpendiculars

meet the reference triangle at the contact

points. The lines containing the contact

points and the opposite vertices on the

reference triangle coincide at the Gergonne

point (see Figure 3).

Importantly, the contact points have

special meaning to the kissing coins

problem. Given three noncollinear points,

one can uniquely construct the three

mutually tangent circles centered at these

points. Construct the reference triangle with

these vertices and its contact triangle, as

described above; the mutually tangent

circles are centered at the vertices of the

reference triangle and contain the nearby

vertices of the contact triangle! Hence,

three non-collinear points uniquely

determine the radii of the kissing coins.

To construct the Gergonne line, one

must understand the notion of triangles in

perspective. The idea of perspective was

introduced in the Renaissance to create the

idea of depth in art. The principle of

perspective is that all lines meet at a point,

thus providing depth. A well-known example

of this is Leonardo da Vinci’s Last Supper,

where all the lines forming the walls, ceiling

and edges of the table meet at a point of

perspective, which happens to be the head

of Christ.

The idea of perspective objects can

be applied in geometry in relation to

triangles. The two triangles ∆ABC and

∆A'B'C' in Figure 4 are perspective from a

line since the extensions of their three pairs

of corresponding sides meet in collinear

points X, Y, and Z. The line joining these

points is called the perspectrix. It can also

be said that two triangles are perspective

from a point if their three pairs of lines

joining their corresponding sides meet at a

point of concurrence O. This point is called

the

perspector,

perspective center,

homology center, or pole [9, Perspective

Triangles].

In the kissing coins problem the

triangle formed by the centers of the circles

and its contact triangle are perspective

triangles. The perspector and the

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Figure 4. Triangles in perspective: Triangles ∆ABC and ∆A'B'C' are perspective from a line since

the extensions of their three pairs of corresponding sides meet in collinear points X, Y, and Z.

perspectrix of these two triangles are called

the Gergonne point and Gergonne line,

respectively, as shown in Figure 5.

The union of the Euler line, Soddy

line and Gergonne line form the EGST, as

seen in Figure 6. The vertices of EGST are

known as the de Longchamps point, Evans

point and Fletcher point. The Soddy line

and Gergonne line always form a right angle

at the Fletcher point [8, p. 328]. As

mentioned previously the Soddy points,

being the centers of the inner and outer

Soddy circles, lie on the Soddy line and form

a harmonic range with the incenter and

Gergonne point [8, p. 326]. Further, the

radii of both Soddy circles can be expressed

in terms of ratios of the radii of the three

given circles and the incircle. We will see in

the next section that two of the solution

circles to Apollonius’ problem are centered

on the Soddy line of the EGST.

Figure 5. Gergonne point and line.

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Figure 6. Euler-Gergonne-Soddy triangle.

III.

THE EPPSTEIN SOLUTION

David Eppstein, a professor in the

Department of Information and Computer

Science at the University of California at

Irvine, is a researcher in the areas of

computational geometry and graph

algorithms. Eppstein is also the founder and

author of the popular Geometry Junkyard

website. On this site Eppstein details a

solution, discovered by Eppstein himself,

that constructs Frederick Soddy’s circles

with a compass and straightedge. The basis

of Eppstein’s solution comes from an article

in The American Mathematical Monthly [13].

In this article Eppstein proves that three

lines through opposite points of tangency of

any four mutually tangent spheres in three-

space are coincident. A resulting corollary

from this lemma is that three lines through

opposite points of tangency of any four

mutually tangent circles in the plane are

coincident.

Eppstein’s solution is as follows

(see Figure 7a): form a triangle connecting

the three circle centers and drop a

perpendicular line from each center to the

opposite triangle edge. Each of these lines

cuts its circle at two points. Seen in Figure

7b, construct a line from each cut point to

the point of tangency of the other two

circles. These lines cut their circles in two

more points, yielding six total, which are the

points of tangency of the Soddy circles.

Once these six points are known, the Soddy

circles’ centers are easily found to lie on the

line determined by the incenter and

Gergonne point, a.k.a. the Soddy line.

In Eppstein’s solution, one may

notice that there are two sets of three lines,

as seen in Figure 7b, each intersecting at a

common point. This is the result of the

corollary from Eppstein’s article mentioned

above. Eppstein points out that despite their

simplicity of definition and the large amount

of study into triangle geometry, these two

points do not appear in the list of over 1,000

known triangle centers collected by Clark

Kimberling and Peter Yff [13, p. 65]. Thus,

these two points have become known as the

Eppstein points and, remarkably, they lie on

the Soddy line and form a harmonic range

with the Gergonne point and incenter [8, p.

327].

Figure 7a. The Eppstein solution.

Figure 7b. Soddy circle and Eppstein

points.

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Figure 8. Inversion of line l through circle C.

IV.

THE GERGONNE SOLUTION

Joseph Diaz Gergonne was born in

France in 1771 and died there in 1859. He

spent most of his youth serving in the

military until 1795. Afterwards, he began his

mathematical study, which spawned a

number of mathematical ideas but mostly

focused on the area of geometry. Gergonne

is well known for creating the journal

officially called the Annales de

Mathématique Pures et Appliquées but

became known as Annales de Gergonne

[14, p. 226]. His journal featured such

prominent mathematicians as Jakob Steiner

and Evariste Galois. Mentioned earlier are a

few of the results of his work in the solution

of the Soddy circles. Not only did Gergonne

supply all eight solution circles to Apollonius’

problem, he also introduced the word polar

as it applies to inversion geometry.

Inversion geometry deals with

transformations of the plane that leave a

given circle fixed while taking its interior

points to its exterior and vice versa. Not just

the subject of advanced Euclidean

geometry, inversion geometry arises in

hyperbolic geometry and conformal

mappings of the complex plane. While the

details and intricacies of inversion geometry

are beyond the scope of this paper, we do

make use of the fact that we can invert any

point through a given circle using compass

and straightedge constructions [15].

For our purposes, we need the

inversion geometry fact that a circle

inversion through the circle C of radius r

centered O in Figure 8 takes the line l to the

circle with diameter

. In this

arrangement P and Q are on the line

perpendicular to l containing O, and

(OP)(OQ) = r

. Points P and Q are easily

constructed with compass and straightedge,

and point Q is called the inversion pole of

the line l.

Gergonne’s solution requires

constructing the dilation points for each pair

of circles [16]. The dilation points of a pair

of circles are the two points of central

similarity about which one circle can be

dilated (or contracted) to the other. As there

are three pairs of circles with two dilation

points each, this process yields six points.

These lie three by three on four lines,

forming a four-line geometry, as illustrated in

Figure 9.

Determine the inversion poles of

one of the dilation lines with respect to each

of the three circles, as in Figure 10a, and

connect the inversion poles with the radical

center, as shown in Figure 10b. The radical

center is the intersection of the three radical

axes; the radical axis of two circles is the

line that contains the center points of all

Figure 9. Dilation points and lines

Figure 10a. Inversion poles

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Figure 10b. Tangent points.

Figure 11a. Tangent circle.

Figure 11b. Another tangent circle.

that are orthogonal to both of the given

circles. In this case, each line containing the

radical center and an inversion pole

intersects its respective circle at two points.

The center of the upper circle in

Figure 10b is alone on one side of the

dilation line formed by the dilation points. On

that circle pick the intersection point furthest

from the radical center. On the other two

circles pick the near intersection points.

These three points are the tangent points for

the solution circle shown in Figure 11a. The

other three intersection points are the

tangent points for another solution circle,

shown in Figure 11b.

This construction sequence yields

all four lines formed from the dilation points,

and for each line produces at most two

solution circles. Thus, all eight solution

circles can be constructed. We should

mention that this process, while not terribly

complicated, does require careful record

keeping. When constructing Gergonne’s

solution, we used the interactive geometry

software program Cinderella™.

Cinderella™ allows one to hide lines and

circles. Without this capability we do not

believe that we could have constructed the

solution. As pictured below in Figure 12, the

solution becomes very cluttered when every

line and circle is visible. We are truly

impressed that Gergonne, or anyone before

the computer age, could have the patience

to perform this feat.

V. CONNECTIONS

AND

EXTENSIONS

In this section we observe some

connections relating the EGST, Eppstein’s

solution and Gergonne’s solution and

suggest some routes for further study.

One of the most intriguing properties

that we have found is the relation of the

Gergonne line and the Gergonne solution.

Since both share the name of J. D.

Gergonne it might come as no surprise that

they are related, but we do not consider this

fact to be obvious. In Gergonne’s solution

one may recall that a line formed by the

three outer dilation points, which we call the

outer dilation line, is used to find two

solution circles. In the special case where

all three circles are mutually tangent, the

Gergonne line and the outer dilation line

coincide. Also, when the given circles are

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Figure 12. The Complete Gergonne Solution.

mutually tangent, the two solution circles

formed by the outer dilation line are the

Soddy circles. Thus, one begins to see the

importance of the Euler-Gergonne-Soddy

triangle and the role it plays in Apollonius’

problem. Furthermore, if the three given

circles are mutually tangent, the incenter of

the reference triangle coincides with the

radical center, and the radical axes that

define the radical center are the

perpendicular lines from the triangle sides to

the incenter.

Until now we have allowed the

reader to believe that Eppstein’s solution to

the kissing coins problem only solves for two

of the solution circles whereas Gergonne’s

solution finds all eight solution circles to the

general Apollonius’ problem. The observant

reader should ask: “Where are the

remaining six tangent circles in the kissing

coins arrangement?” It turns out that each

of the three given circles represents two

solutions. This can best be observed using

geometry software to execute Gergonne’s

solution and arranging the three given

circles to be nearly tangent. One will

observe that the coins, now separated by a

small distance from each other, are each

internally tangent to two solution circles just

slightly larger than the coin itself.

As another special case of three

circles in general position, consider what

happens when the radius of one of the given

circles tends toward zero. Here, the eight

solution circles collapse to four, and they do

so in pairs. When the radius of a second

circle decreases, the four solution circles

collapse to two. Finally, when the radius of

the third circle shrinks, the two solutions

collapse to one. Figure 13 illustrates this

nicely for a sequence of circles whose radii

tend toward zero. Hence, Euclid’s

Proposition 5, Book IV, for finding the

circumcenter of a triangle should really be

viewed as a very special case of Gergonne’s

solution to Apollonius’ problem!

As yet another special case,

consider when the given circles are the

excircles of a triangle. Here, the vertices of

the triangle are the inner dilation points for

the pairs of circles. Surprisingly, the lines

extending the triangle sides are three of the

Apollonius solutions; they are limiting cases

as the radii of three of the solution circles

tend to infinity! Feuerbach’s Theorem

guarantees that the nine-point circle is

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Figure 13. Shrinking circles and collapsing solutions.

simultaneously tangent to the three

excircles, providing a fourth solution circle

[17,p. 46 ]. This solution is analogous to the

inner Soddy circle solution to the kissing

coins problem in that the three excircles are

externally tangent to the nine-point circle. A

fifth solution circle lies so that the excircles

are internally tangent, and the remaining

three solution circles each have one excircle

internally tangent and the other two excircles

externally tangent.

With regard to further study, the

interested reader may wish to pursue the

following topics.

In Apollonius’ problem the three

given objects are taken from among circles,

points and lines. The latter two objects

should be thought of as limiting cases when

the radius of a circle approaches zero or

infinity. Gergonne’s solution only applies to

the case when all three objects are circles

since circle inversion is used. Rather than

use three circles, one may wish to examine

all combinations of points, lines and circles,

decide for which configurations all eight

solution circles appear, and use geometry

software to construct solution circles. For

example, no solution circles exist in the case

of three parallel lines, and two solution

circles exist for two parallel lines cut by a

transversal. Are there any connections to

Gergonne’s solution?

It is also worth exploring the special

case of three circles centered at points that

are the vertices of an isosceles or an

equilateral triangle. If at least two of the

three sides of the reference triangle are

congruent, the Euler line and the Soddy line

coincide, so the EGST is degenerate. In the

context of Eppstein’s solution, this

corresponds to kissing coins with equal radii.

Given three noncollinear points,

consider all sets of three non-overlapping

circles centered at these points. What are

the possible locations for the centers of the

tangency circles for which the three circles

are either all internally tangent or all

externally tangent? The kissing coins

problem is the special case when the three

circles centered at the triangle vertices are

tangent to each other. In this instance,

perhaps the inner and outer Soddy centers

that solve this problem are known triangle

centers.

Finally, as noted in section 2, the EGST

is always a right triangle. A benefit of

Cinderella™ is the ability to dynamically

move elements while preserving the incident

relations among points, lines and circles.

With this capability the vertices of the

original triangle can be moved freely, and

we have observed that the EGST always

appears to be long and skinny (see Figure

6). The maximum value of the angle

between the Euler line and Gergonne line is

90 degrees, but what is the smallest

possible value of this angle?

REFERENCES

1. Boyer, C.

B. A History of Mathematics.

New York: John Wiley & Sons, 1968.

2. Boyer, C. B.; revised by Merzbach, Uta.

A History of Mathematics. New York:

John Wiley & Sons, 1989.

3. Cajori, Florian. A History of

Mathematics. New York: Chelsea

Publishing Company, 1980.

4. Gow, James. A Short History of Greek

Mathematics. New York: G. E. Stechert

& Co., 1923.

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

5. Joyce, David E., “Euclid’s Elements.”

http://aleph0.clarku.edu/~djoyce/java/ele

ments/elements.html.

6. Heath, Thomas. A History of Greek

Mathematics. New York: Oxford

University Press, Vol. 2, 1921.

7. Court, N. A. “Historically Speaking, -

The Problem of Apollonius.” The

Mathematics Teacher. October 1961,

pp. 444-452.

8. Oldknow, A. “The Euler-Gergonne-

Soddy Triangle of a Triangle.” The

American Mathematical Monthly 103

(1996), pp. 319-329.

9. Weisstein, Eric W. Eric Weisstein’s

World of Mathematics.

http://mathworld.wolfram.com.

10. Soddy, F. “The Kiss Precise.” Nature

137 (1936), p. 1021.

11.

Eppstein, David. “Tangencies:

Apollonian Circles.” Geometry Junkyard.

www.ics.uci.edu/~eppstein/junkyard/tan

gencies/apollonian.html.

12.

Bogomolny, Alexander. “Apollonius

Problem: What is it?” Cut The Knot!

www.cut-the-

knot.com/Curriculum/Geometry/Apolloni

us.shtml.

13. Eppstein, David. “Tangent Spheres and

Triangle Centers.” The American

Mathematical Monthly 108 (2001), pp.

63-66.

14.

Boyer, C. B. History of Analytic

Geometry. New York: Scripta

Mathematica, 1956.

15. Goodman-Strauss, Chaim. “Compass

and Straightedge in the Poincare Disk.”

The American Mathematical Monthly

108 (2001), pp. 38-49.

16.

Kunkel, Paul. “Tangent Circles.”

http://whistleralley.com/tangents/tangent

s.htm.

17. Baragar, Arthur. A Survey of Classical

and Modern Geometries with Computer

Activities. New Jersey: Prentice-Hall Inc.

2001.

APPENDIX: Glossary of Terms

Points and Centers

Centroid

The intersection of the medians

Circumcenter The

intersection

of the perpendicular bisectors

Incenter

Intersection of the angle bisectors

Orthocenter

Intersection of the altitudes

Gergonne point

Perspector of a reference triangle and its contact triangle

Evans point

Intersection of the Euler and Soddy lines

Fletcher point

Intersection of the Soddy and Gergonne lines

De Longchamps

Intersection of the Euler and Gergonne lines

Lines

Euler line

Defined by the circumcenter and orthocenter; also contains the

centroid and nine-point center

Gergonne line

Line of perspective for a reference triangle and its contact triangle

Soddy line

Defined by the incenter and Gergonne point; also contains the

Eppstein and Soddy points

Circles

Circumcircle

Unique circle containing a triangle’s vertices

Incircle

Unique circle internally tangent to all three triangle sides

Nine-point circle

Contains side midpoints, feet of altitudes, and midpoints of

segments joining the orthocenter to the vertices

Inner Soddy circle

Circle that is internally tangent to the kissing coins

Outer Soddy circle

Circle that is externally tangent to the kissing coins

Triangles

Contact triangle

Defined by the points of tangency of reference triangle and its

incircle; vertices are where the kissing coins touch

Euler-Gergonne-Soddy

triangle

Triangle formed by the Euler, Gergonne and Soddy lines

AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH

VOL. 3 NO. 1 (2004)

Document Outline

Apollonius’ Problem: A Study of Solutions and The
- - David Gisch and Jason M. Ribando
    - University Northern Iowa
  - The Kiss Precise

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