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AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 3 NO. 1 (2004)

Apollonius Problem: A Study of Solutions andTheir Connections

David Gisch and Jason M. RibandoDepartment of Mathematics

University Northern IowaCedar 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 givencircles. More generally, Apollonius' problem asks to construct the circle which is tangent to anythree objects that may be any combination of points, lines, and circles. The case when all threeobjects are circles is the most complicated case since up to eight solution circles are possibledepending on the arrangement of the given circles. Within the last two centuries, solutions havebeen given by J. D. Gergonne in 1816, by Frederick Soddy in 1936, and most recently by DavidEppstein in 2001. In this report, we illustrate the solution using the geometry softwareCinderella, survey some connections among the three solutions, and provide a framework forfurther study.

I. INTRODUCTION

Apollonius of Perga was known as'The Great Geometer'. He should not beconfused with other Greek scholars namedApollonius, for it was a common name.Little is known of his life except that he wasborn in Perga, Pamphylia, which today isknown as Murtina, or Murtana, and is now inAntalya, Turkey. The years 262 to 190 B.C.have been suggested for his life [1-3]. It iscommonly believed that Apollonius went toAlexandria where he studied under thefollowers of Euclid and possibly taught therelater.

This paper focuses on a problemsolved by Apollonius in his book

Tangencies. Apollonius works have had agreat influence on the development ofmathematics [4]. In particular, his famousbook Conics introduced terms which arefamiliar to us today such as parabola, ellipseand hyperbola. In Book IV of the Elements,Euclid details how to construct a circletangent 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 latterconstruction is accomplished by finding theintersection point of the perpendicularbisectors of any two sides of the trianglewith the three given points as vertices. InTangencies Apollonius poses ageneralization to Euclids two propositions:given any three points, lines or circles in theplane, construct a circle which contains thepoints and is tangent to the lines and circles.Apollonius enumerated the ten combinationsof points, lines and circles and solved thecases not already solved by Euclid [6,p.182]. The case when all three objects arecircles is the most complicated of the ten

cases since up to eight solution circles arepossible depending on the arrangement ofthe circles (see Figure 1).

Since no copy of ApolloniusTangencies has survived the ages, Pappusof Alexandria deserves credit for eternallylinking Apollonius name with the tangentsproblem. Pappus, who lived some fivecenturies after Apollonius and is known

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Figure 1. Eight solution circles to Apollonius problem.

mostly for his encyclopedic recording andcommenting of Greek mathematics, wrote aTreasury of Analysis in Book VII of hisMathematical Collections dedicated to worksof Euclid, Aristaeus, and Apollonius. Herehe succinctly states Apollonius problem,acknowledges the ten cases, and provides acompass and straightedge solution for atleast one solution circle [6, p. 182].

Excepting Arabic reconstructions ofApollonius works, Apollonius problem laydormant in the literature until Franois Vite

(1540-1603) restored the Tangencies in1603. Having unveiled Apollonius solution,Vite challenged Adrianus Romanus to drawa circle tangent to three given circles butwas disappointed by Romanus use ofconics. A century later Newton also wentbeyond compass and straightedge solutionsby employing hyperbolas [7]. Agreeing withVites preference for compass andstraightedge solutions, we are motivated toinclude three related constructions under thesame cover: the Euler-Gergonne-Soddytriangle, which contains the centers of thetwo solution circles in the special case when

the circles are mutually tangent or kissing;a solution found by David Eppstein in 2001for the same special case; and anadaptation to Gergonnes analytic solutionwhich constructs all eight solution circles inthe general case for three circles in genericposition. We conclude with someconnections among the three solutions and

provide a framework for further studybeyond the scope of this paper.

II. THE EULER-GERGONNE-SODDYTRIANGLE OF A TRIANGLE

A special case of Apollonius'problem is known today as the three coinsproblem, or kissing coins problem. In thisvariant, the three circles, of possibly differentradii, are taken to be mutually tangent.There are two solutions to this special caseof Apollonius problem: a small circle where

all three given circles are externally tangent,and a large circle where the three givencircles are internally tangent. In 1643 RenDescartes sent a letter to Princess Elisabethof Bohemia in which he provided a solutionto this special case of Apollonius problem.His solution became known as Descartescircle theorem. Philip Beecroft, an Englishamateur mathematician, rediscoveredDescartes circle theorem in 1842. Then itwas discovered again in 1936 by FrederickSoddy (1877-1956), who had won a NobelPrize in 1921 for his discovery of isotopes

[8]. Soddy expressed the theorem in theform of a poem, "The Kiss Precise," whichwas published in the journal Nature and isincluded below. It may have been the flavorof the added poem that set Soddy apartfrom his predecessors, as the two circles areknown today as the innerand outer Soddycircles. Additionally, Soddy extended thetheorem to the analogous formula for six

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AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 3 NO. 1 (2004)spheres in three dimensions [9, SoddyCircles].

Though Soddy provided an analyticsolution to the kissing coins problem, we donot know whether he constructed a syntheticsolution using compass and straightedge.For our part we find compass andstraightedge solutions easier to understandthan their analytic counterparts. Thissection is devoted to the Euler-Gergonne-Soddy triangle (EGST). Though it does notprovide a solution to the kissing coinsproblem, it does provide a unique insight tofinding the Soddy circles. Many of theconnections between the EGST and theSoddy circles solution to the kissing coinsproblem were inspired by Oldknows article[8], which uses trilinear coordinates,harmonic ranges, and parameterizations oflines to construct the EGST. It is interesting

to note that Oldknow attributes many of hisinvestigations to the use of geometricsoftware packages, a current trend amonggeometry researchers.

As noted already, the kissing coinsproblem is the special case of Apolloniusproblem where all three circles are mutuallytangent. In this arrangement there are twosolution circles, the inner and outer Soddycircles as shown in Figure 2. There are twoquestions of interest that arise from thisspecial case: what are the radii of the Soddycircles and where are their centers located?

As a partial answer to the latterquestion, the Soddy centers lie on the line

Figure 2. Soddy circles.

The Kiss Precise

For pairs of lips to kiss maybeInvolves no trigonometry.'Tis not so when four circles kissEach one the other three.

To bring this off the four must beAs three in one or one in three.If one in three, beyond a doubtEach gets three kisses from without.If three in one, then is that oneThrice kissed internally.

Four circles to the kissing come.The smaller are the benter.The bend is just the inverse ofThe 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 lineAnd concave bends have minus sign,*The sum of the squares of all four bendsIs half the square of their sum.*

To spy out spherical affairsAn oscular surveyorMight find the task laborious,The sphere is much the gayer,And now besides the pair of pairsA 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 Gergonnepointof the reference triangle. This line isknown as the Soddy line and it contains oneof the sides of the EGST. Given threevertices of a triangle, we show how toconstruct its EGST. The reader may want torefer to the Appendix for a glossary ofgeometric terms involved in thisconstruction.

We start with the Euler line since itis the easiest to construct. Given areference triangle, find the circumcenterasthe intersection of the perpendicularbisectors and the orthocenter as theintersection of the altitudes. These twopoints determine the Euler line, which also

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

contains the triangles centroid (theintersection of the medians). The Euler linealso contains a number of other importanttriangle centers including the center of thenine-point circle.

The Soddy line is determined by theincenter, where the angle bisectors coincide,and the Gergonne point. The incenteris thecenter of the unique circle that is internallytangent to its reference triangle at three

points. These three points are called thecontact points, and together they form thecontact triangle of the reference triangle.The Gergonne point can most easily befound as follows: erect perpendicularscontaining the incenter from each side of thereference triangle. These perpendicularsmeet the reference triangle at the contactpoints. The lines containing the contactpoints and the opposite vertices on thereference triangle coincide at the Gergonnepoint (see Figure 3).

Importantly, the contact points have

special meaning to the kissing coinsproblem. Given three noncollinear points,one can uniquely construct the threemutually tangent circles centered at thesepoints. Construct the reference triangle withthese vertices and its contact triangle, asdescribed above; the mutually tangentcircles are centered at the vertices of thereference triangle and contain the nearbyvertices of the contact triangle! Hence,

three non-collinear points uniquelydetermine the radii of the kissing coins.

To construct the Gergonne line, onemust understand the notion of triangles in

perspective. The idea of perspective wasintroduced in the Renaissance to create theidea of depth in art. The principle ofperspective is that all lines meet at a point,thus providing depth. A well-known exampleof this is Leonardo da Vincis Last Supper,

where all the lines forming the walls, ceilingand edges of the table meet at a point ofperspective, which happens to be the headof Christ.

The idea of perspective objects canbe applied in geometry in relation totriangles. The two triangles ABC and

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

line since the extensions of their three pairsof corresponding sides meet in collinearpoints X, Y, and Z. The line joining thesepoints is called the perspectrix. It can alsobe said that two triangles are perspective

from a point if their three pairs of lines joining their corresponding sides meet at apoint of concurrence O. This point is calledthe perspector, perspective center,homology center, or pole [9, PerspectiveTriangles].

In the kissing coins problem thetriangle formed by the centers of the circlesand its contact triangle are perspectivetriangles. The perspector and the

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Figure 4. Triangles in perspective: Triangles ABCand A'B'C'areperspective from a line since

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

perspectrix of these two triangles are calledthe Gergonne point and Gergonne line,respectively, as shown in Figure 5.

The union of the Euler line, Soddyline and Gergonne line form the EGST, asseen in Figure 6. The vertices of EGST areknown as the de Longchamps point, Evans

point and Fletcher point. The Soddy lineand Gergonne line always form a right angleat the Fletcher point [8, p. 328]. Asmentioned previously the Soddy points,

being the centers of the inner and outerSoddy circles, lie on the Soddy line and forma harmonic range with the incenter andGergonne point [8, p. 326]. Further, theradii of both Soddy circles can be expressedin terms of ratios of the radii of the threegiven circles and the incircle. We will see inthe next section that two of the solutioncircles to Apollonius problem are centeredon the Soddy line of the EGST.

Figure 5. Gergonne point and line.

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Figure 6. Euler-Gergonne-Soddy triangle.

III. THE EPPSTEIN SOLUTION

David Eppstein, a professor in theDepartment of Information and ComputerScience at the University of California atIrvine, is a researcher in the areas ofcomputational geometry and graphalgorithms. Eppstein is also the founder andauthor of the popular Geometry Junkyardwebsite. On this site Eppstein details asolution, discovered by Eppstein himself,that constructs Frederick Soddys circleswith a compass and straightedge. The basisof Eppsteins solution comes from an article

in The American Mathematical Monthly[13].In this article Eppstein proves that threelines through opposite points of tangency ofany four mutually tangent spheres in three-space are coincident. A resulting corollaryfrom this lemma is that three lines throughopposite points of tangency of any fourmutually tangent circles in the plane arecoincident.

Eppsteins solution is as follows(see Figure 7a): form a triangle connectingthe three circle centers and drop aperpendicular line from each center to the

opposite triangle edge. Each of these linescuts its circle at two points. Seen in Figure7b, construct a line from each cut point tothe point of tangency of the other twocircles. These lines cut their circles in twomore points, yielding six total, which are thepoints of tangency of the Soddy circles.Once these six points are known, the Soddycircles centers are easily found to lie on the

line determined by the incenter andGergonne point, a.k.a. the Soddy line.

In Eppsteins solution, one maynotice that there are two sets of three lines,as seen in Figure 7b, each intersecting at acommon point. This is the result of thecorollary from Eppsteins article mentionedabove. Eppstein points out that despite theirsimplicity of definition and the large amountof study into triangle geometry, these twopoints do not appear in the list of over 1,000known triangle centers collected by ClarkKimberling and Peter Yff [13, p. 65]. Thus,these two points have become known as theEppstein points and, remarkably, they lie onthe Soddy line and form a harmonic rangewith the Gergonne point and incenter [8, p.327].

Figure 7a. The Eppstein solution.

Figure 7b. Soddy circle and Eppsteinpoints.

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Figure 8. Inversion of line l through circle C.

IV. THE GERGONNE SOLUTION

Joseph Diaz Gergonne was born inFrance in 1771 and died there in 1859. Hespent most of his youth serving in the

military until 1795. Afterwards, he began hismathematical study, which spawned anumber of mathematical ideas but mostlyfocused on the area of geometry. Gergonneis well known for creating the journalofficially called the Annales deMathmatique Pures et Appliques butbecame known as Annales de Gergonne[14, p. 226]. His journal featured suchprominent mathematicians as Jakob Steinerand Evariste Galois. Mentioned earlier are afew of the results of his work in the solutionof the Soddy circles. Not only did Gergonnesupply all eight solution circles to Apolloniusproblem, he also introduced the word polaras it applies to inversion geometry.

Inversion geometry deals withtransformations of the plane that leave agiven circle fixed while taking its interiorpoints to its exterior and vice versa. Not justthe subject of advanced Euclideangeometry, inversion geometry arises inhyperbolic geometry and conformalmappings of the complex plane. While thedetails and intricacies of inversion geometryare beyond the scope of this paper, we domake use of the fact that we can invert any

point through a given circle using compassand straightedge constructions [15].

For our purposes, we need theinversion geometry fact that a circleinversion through the circle C of radius rcentered O in Figure 8 takes the line l to the

circle with diameter OQ . In this

arrangement P and Q are on the lineperpendicular to l containing O, and

(OP)(OQ) = r2. Points P and Q are easily

constructed with compass and straightedge,and point Q is called the inversion pole ofthe line l.

Gergonnes solution requiresconstructing the dilation points for each pairof circles [16]. The dilation points of a pairof circles are the two points of centralsimilarity about which one circle can bedilated (or contracted) to the other. As thereare three pairs of circles with two dilationpoints each, this process yields six points.These lie three by three on four lines,forming a four-line geometry, as illustrated inFigure 9.

Determine the inversion poles ofone of the dilation lines with respect to eachof the three circles, as in Figure 10a, andconnect the inversion poles with the radicalcenter, as shown in Figure 10b. The radical

center is the intersection of the three radicalaxes; the radical axis of two circles is theline that contains the center points of all

Figure 9. Dilation points and lines

Figure 10a. Inversion poles

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Figure 10b. Tangent points.

Figure 11a. Tangent circle.

Figure 11b. Another tangent circle.

that are orthogonal to both of the givencircles. In this case, each line containing theradical center and an inversion poleintersects its respective circle at two points.

The center of the upper circle inFigure 10b is alone on one side of thedilation line formed by the dilation points. Onthat circle pick the intersection point furthestfrom the radical center. On the other twocircles pick the near intersection points.These three points are the tangent points forthe solution circle shown in Figure 11a. Theother three intersection points are thetangent points for another solution circle,shown in Figure 11b.

This construction sequence yieldsall four lines formed from the dilation points,and for each line produces at most twosolution circles. Thus, all eight solutioncircles can be constructed. We should

mention that this process, while not terriblycomplicated, does require careful recordkeeping. When constructing Gergonnessolution, we used the interactive geometrysoftware program Cinderella.Cinderella allows one to hide lines andcircles. Without this capability we do notbelieve that we could have constructed thesolution. As pictured below in Figure 12, thesolution becomes very cluttered when everyline and circle is visible. We are trulyimpressed that Gergonne, or anyone beforethe computer age, could have the patience

to perform this feat.

V. CONNECTIONS ANDEXTENSIONS

In this section we observe someconnections relating the EGST, Eppsteinssolution and Gergonnes solution andsuggest some routes for further study.

One of the most intriguing propertiesthat we have found is the relation of theGergonne line and the Gergonne solution.Since both share the name of J. D.

Gergonne it might come as no surprise thatthey are related, but we do not consider thisfact to be obvious. In Gergonnes solutionone may recall that a line formed by thethree outer dilation points, which we call theouter dilation line, is used to find twosolution circles. In the special case whereall three circles are mutually tangent, theGergonne line and the outer dilation linecoincide. Also, when the given circles are

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Figure 12. The Complete Gergonne Solution.

mutually tangent, the two solution circlesformed by the outer dilation line are theSoddy circles. Thus, one begins to see theimportance of the Euler-Gergonne-Soddytriangle and the role it plays in Apolloniusproblem. Furthermore, if the three givencircles are mutually tangent, the incenter ofthe reference triangle coincides with theradical center, and the radical axes thatdefine the radical center are theperpendicular lines from the triangle sides tothe incenter.

Until now we have allowed thereader to believe that Eppsteins solution tothe kissing coins problem only solves for twoof the solution circles whereas Gergonnessolution finds all eight solution circles to thegeneral Apollonius problem. The observantreader should ask: Where are the

remaining six tangent circles in the kissingcoins arrangement? It turns out that eachof the three given circles represents twosolutions. This can best be observed usinggeometry software to execute Gergonnessolution and arranging the three givencircles to be nearly tangent. One willobserve that the coins, now separated by asmall distance from each other, are each

internally tangent to two solution circles justslightly larger than the coin itself.

As another special case of threecircles in general position, consider whathappens when the radius of one of the givencircles tends toward zero. Here, the eightsolution circles collapse to four, and they doso in pairs. When the radius of a secondcircle decreases, the four solution circlescollapse to two. Finally, when the radius ofthe third circle shrinks, the two solutionscollapse to one. Figure 13 illustrates thisnicely for a sequence of circles whose radiitend toward zero. Hence, EuclidsProposition 5, Book IV, for finding thecircumcenter of a triangle should really beviewed as a very special case of Gergonnessolution to Apollonius problem!

As yet another special case,

consider when the given circles are theexcircles of a triangle. Here, the vertices ofthe triangle are the inner dilation points forthe pairs of circles. Surprisingly, the linesextending the triangle sides are three of theApollonius solutions; they are limiting casesas the radii of three of the solution circlestend to infinity! Feuerbachs Theoremguarantees that the nine-point circle is

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Figure 13. Shrinking circles and collapsing solutions.

simultaneously tangent to the threeexcircles, providing a fourth solution circle[17,p. 46 ]. This solution is analogous to theinner Soddy circle solution to the kissingcoins problem in that the three excircles are

externally tangent to the nine-point circle. Afifth solution circle lies so that the excirclesare internally tangent, and the remainingthree solution circles each have one excircleinternally tangent and the other two excirclesexternally tangent.

With regard to further study, theinterested reader may wish to pursue thefollowing topics.

In Apollonius problem the threegiven objects are taken from among circles,points and lines. The latter two objectsshould be thought of as limiting cases whenthe radius of a circle approaches zero orinfinity. Gergonnes solution only applies tothe case when all three objects are circlessince circle inversion is used. Rather thanuse three circles, one may wish to examineall combinations of points, lines and circles,decide for which configurations all eightsolution circles appear, and use geometrysoftware to construct solution circles. Forexample, no solution circles exist in the caseof three parallel lines, and two solutioncircles exist for two parallel lines cut by atransversal. Are there any connections toGergonnes solution?

It is also worth exploring the specialcase of three circles centered at points thatare the vertices of an isosceles or anequilateral triangle. If at least two of thethree sides of the reference triangle arecongruent, the Euler line and the Soddy linecoincide, so the EGST is degenerate. In thecontext of Eppsteins solution, thiscorresponds to kissing coins with equal radii.

Given three noncollinear points,consider all sets of three non-overlappingcircles centered at these points. What arethe possible locations for the centers of thetangency circles for which the three circles

are either all internally tangent or allexternally tangent? The kissing coinsproblem is the special case when the threecircles centered at the triangle vertices aretangent to each other. In this instance,perhaps the inner and outer Soddy centersthat solve this problem are known trianglecenters.

Finally, as noted in section 2, the EGSTis always a right triangle. A benefit ofCinderella is the ability to dynamicallymove elements while preserving the incidentrelations among points, lines and circles.With this capability the vertices of theoriginal triangle can be moved freely, andwe have observed that the EGST alwaysappears to be long and skinny (see Figure6). The maximum value of the anglebetween the Euler line and Gergonne line is90 degrees, but what is the smallestpossible 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 ofMathematics. New York: ChelseaPublishing Company, 1980.

4. Gow, James. A Short History of GreekMathematics. New York: G. E. Stechert& Co., 1923.

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AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 3 NO. 1 (2004)5. Joyce, David E., Euclids Elements.

http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.

6. Heath, Thomas. A History of GreekMathematics. New York: OxfordUniversity Press, Vol. 2, 1921.

7. Court, N. A. Historically Speaking, -The Problem of Apollonius. TheMathematics Teacher. October 1961,pp. 444-452.

8. Oldknow, A. The Euler-Gergonne-Soddy Triangle of a Triangle. The

American Mathematical Monthly103(1996), pp. 319-329.

9. Weisstein, Eric W. Eric WeissteinsWorld of Mathematics.http://mathworld.wolfram.com.

10. Soddy, F. The Kiss Precise. Nature137 (1936), p. 1021.

11. Eppstein, David. Tangencies:

Apollonian Circles. Geometry Junkyard.www.ics.uci.edu/~eppstein/junkyard/tangencies/apollonian.html.

12. Bogomolny, Alexander. ApolloniusProblem: What is it? Cut The Knot!www.cut-the-knot.com/Curriculum/Geometry/Apollonius.shtml.

13. Eppstein, David. Tangent Spheres andTriangle Centers. The AmericanMathematical Monthly 108 (2001), pp.63-66.

14. Boyer, C. B. History of AnalyticGeometry. New York: ScriptaMathematica, 1956.

15. Goodman-Strauss, Chaim. Compassand Straightedge in the Poincare Disk.The American Mathematical Monthly108 (2001), pp. 38-49.

16. Kunkel, Paul. Tangent Circles.http://whistleralley.com/tangents/tangents.htm.

17. Baragar, Arthur. A Survey of Classical

and Modern Geometries with ComputerActivities. 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 linesFletcher 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 thecentroid 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 theEppstein and Soddy points

Circles

Circumcircle Unique circle containing a triangles vertices

Incircle Unique circle internally tangent to all three triangle sides

Nine-point circle Contains side midpoints, feet of altitudes, and midpoints ofsegments 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 itsincircle; vertices are where the kissing coins touch

Euler-Gergonne-Soddytriangle

Triangle formed by the Euler, Gergonne and Soddy lines

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