Invariant Center Power and Elliptic Loci of Poncelet Triangles
aa r X i v : . [ m a t h . M G ] F e b INVARIANT CENTER POWER AND ELLIPTIC LOCIOF PONCELET TRIANGLES
MARK HELMAN, DOMINIQUE LAURAINRONALDO GARCIA, AND DAN REZNIK
Abstract.
We study center power with respect to circles derived from Pon-celet 3-periodics (triangles) in a generic pair of ellipses as well as loci of theirtriangle centers. We show that (i) for any concentric pair, the power of thecenter with respect to either circumcircle or Euler’s circle is invariant, and (ii)if a triangle center of a 3-periodic in a generic nested pair is a fixed linearcombination of barycenter and circumcenter, its locus over the family is anellipse. Introduction
Poncelet N-periodics are families of N-gons inscribed in a first conic while si-multaneously circumscribing a second conic [5]. We continue our study of loci andinvariants of Poncelet 3-periodics (see related work below). Previously we focusedon families interscribed between concentric, axis-aligned ellipse pairs. Here we ex-pand the analysis to a generic pair of nested ellipses and explore (i) the power ofthe center with respect to well-known circles, and (ii) loci of triangle centers undervarious ellipse arrangements, see Figure 1. Recall triangle centers are points in theplane of a triangle (e.g, incenter, circumcenter, etc.) whose trilinear coordinatesobey certain conditions [12].
Date : January, 2021.
Figure 1.
A pair of ellipses in general position which admits a Poncelet 3-periodic family (blue).Let the outer one be centered at the origin O . Their major axes are tilted by θ , and their centersdisplaced by O c = ( x c , y c ) . 1 HELMAN, LAURAIN, GARCIA, AND REZNIK Main Results. • Section 3: We first show that over 3-periodics in several concentric, axis-aligned pairs – confocal, homothetic, with incircle, with circumcircle, andexcentral – the power of the center with respect to either the circumcir-cle or Euler’s circle is invariant. Using analytic geometry (with trilinearcoordinates), we derive explicit formulas for said powers for each family. • Section 4: using CAS-based manipulation, we generalize this by provingthat the power of the center is invariant for 3-periodics for any genericconcentric pair (aligned or not), Theorem 1. • Section 5: We then consider 3-periodics in the non-concentric pair withcircumcircle. Using a special parametrization based on Blaschke products[4], we show that loci of triangle centers which are fixed affine combinationsof the barycenter and circumcenter are circles, whose centers are collinearalong a line passing through the stationary circumcenter. • Section 6 For the generic case of 3-periodics in the non-concentric, non-axis-aligned ellipse pair, we show that triangle centers which are fixed linearcombinations of barycenter and circumcenter will trace out elliptic loci,Theorem 2. These include such centers as the orthocenter, the center ofthe Euler circle, the de Longchamps point, etc.; see Observation 8.In Section 7 we conclude with a few experimental conjectures. Appendix Bcontains a list of symbols used herein.
Related Work.
In [15], the loci of many triangle centers over the poristic family(fixed circumcenter and incenter) are shown to be either stationary, circular, orelliptic. In [20], the loci of vertex and area centroids are proved to be ellipsesover a generic Poncelet family. The circumcenter of mass (which is simply thecircumcenter for Poncelet 3-periodics) is shown to be an ellipse in [22].Properties of 3-periodics in the confocal pair (elliptic billiard) were studied in[18, 11]. A few results and their subsequent proofs include: the elliptic locus of theincenter [19, 7], circumcenter [7, 6], invariant sum of cosines [1, 2], and invariantratio of outer-to-orbit polygon areas [3].In [10] it was shown that over confocal 3-periodics, 29 triangle centers (out of thefirst 100 in [13]) trace out ellipses. Explicit expressions are given for the semi-axesof each locus. In subsequent works, we studied the relationship between (i) poristictriangles and the confocal family (poristic) [9], and (ii) the homothetic family andthe Brocard porism [17], showing that said pairs are images of each other under avariable similarity transform. In [8] we compare several loci and invariants acrossseveral concentric, axis-aligned pairs, grouping them into clusters.2.
Preliminaries
Consider two nested ellipses E and E c with semi-axes ( a, b ) and ( a c , b c ) : E iscentered at the origin O and E c at O c = ( x c , y c ) . Let θ denote the angle betweentheir major axes; see Figure 1. If O c = 0 we call the pair “concentric”. If θ = 0 wecall it “axis-aligned”. Additionally, let c = a − b and c c = a c − b c denote theirhalf focal distances. Note these are the squares of half the focal distance. Definition 1.
The power P X of a point X with respect to a circle centered on C and of radius R is given by [24, Circle Power]: ONCELET CENTER POWER AND LOCI 3 P X ( C, R ) = | X − C | − R Recall the circumcircle passes through triangle vertices. Using Kimberling’snotation for triangle centers [13], let X and R denote its center and radius; theseare known as circumcenter and circumradius. Also recall Euler’s (or Feuerbach’s,or the 9-point) circle: it passes through the sides’ midpoints. Its radius is halfthe circumradius [24, Nine-point circle]. Let X denote its center. The followingshorthands will be used for the power of a point O wrt to either circumcircle orEuler’s circle: P = P O ( X , R ) , P = P O ( X , R/ Concentric, Axis-Aligned: Invariant Power of Origin
In this Section we study the power of the center of the system over severalclassic concentric, axis-aligned Poncelet families. A key observation is that saidpower remains constant with respect to two classic circles (the circumcircle andEuler’s circle), despite the fact that their centers and radii variable.3.1.
Pair with Incircle.
Consider an ellipse pair where the inner ellipse is a circleof radius r ; see Figure 2. The Cayley condition for 3-periodics reduces to r =( ab ) / ( a + b ) [8, Coroll. 1]. By definition, the incenter X lies at the origin and theinradius is constant.Remarkable properties of this family include the fact that (i) it conserves the sumof cosines, (ii) the circumradius R is invariant, and (iii) the locus of both X and X are circles concentric with X . Let r and r denote their radii, respectively.These are given by [8, Section 3]: R = a + b , r = a − b , r = ( a − b ) a + b ) Proposition 1.
Over 3-periodics in the concentric pair with incircle, the power ofthe center O = X with respect to either circumcircle [8, Prop. 1] and Euler’s circleare invariant and given by: P = − ab, P = − ab a + b a + b ) Proof.
We compute powers wrt circumcircle and Euler circle as functions of side-lengths s , s , s : P = − s s s s + s + s P = − ( s + s + s − s ( s + s ) − s ( s + s ) − s ( s + s ) + 4 s s s )4( s + s + s ) P is − /π times the area of the circumellipse centered on the incenter, i.e., P = − ab . In order to simplify P formula we derive the following parametrizationfor the sidelengths: Amongst the first 201 centers in [13] the loci of the following are circles concentric with X : X k , k =
3, 5, 11, 12, 35, 36, 40, 46, 55, 56, 57, 65, 80, 119, 165 [8].
HELMAN, LAURAIN, GARCIA, AND REZNIK
Figure 2.
In the ellipse pair with incircle, the locus of both X and X are concentric circles (redand green). The power of the center O = X wrt to either the circumcircle (dashed red) or Euler’scircle (dashed green) is invariant. s = wτr , s = 4 r ( r + (1 − t ) w )(2 r + (1 − t ) w ) τ + (1 − t ) z , s = 4 r ( r + (1 − t ) w )(2 r + (1 − t ) w ) τ − (1 − t ) z with τ = √ − t and z = p r ( t − w + (1 − t ) w − r , r = ( ab ) / ( a + b ) ,and w = ab . Replacing s , s , s obtain P = r − / w and the formula in theproposition. (cid:3) Pair with Circumcircle.
Consider an ellipse pair where the outer ellipse isa circle of radius a = b = R and the inner one is a concentric ellipse with semi-axes ( a c , b c ) . The Cayley condition for 3-periodics to exist reduces to a c + b c = R [8]. By definition, the incenter X lies at the origin and the circumradius R isconstant. Invariants known to this family include the product of cosines and thesum of sidelengths squared [8]. Referring to Figure 3: Proposition 2.
Over 3-periodics in the concentric pair with circumcircle, the locusof X is a circle concentric with original pair whose radius r is given by r =( a c − b c ) / . Corollary 1.
Over 3-periodics in the concentric pair with circumcircle, the powerof the center O = X with respect to either circumcircle or Euler’s circle is invariantand given by: P = − R , P = r − ( R/ = − a c b c Homothetic Pair.
Consider a pair of concentric, homothetic ellipses admit-ting a 3-periodic family (elliptic billiard); see Figure 4. Amongst the first 201 centers in [13] the loci of the following are circles concentric with X : X k , k =
2, 4, 5, 20, 22, 23, 24, 25, 26, 74, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109,110, 111, 112, 140, 156, 186, 201 [8].
ONCELET CENTER POWER AND LOCI 5
Figure 3.
A circle and a concentric inellipse and a 3-periodic (blue). By definition, X is stationaryat the common center and the circumradius R is constant. The locus of the center X of Euler’scircle (dashed green) is a concentric circle (solid green). Figure 4.
A concentric, homothetic pair of ellipses (black) and a 3-periodic (blue) interscribedbetween them. The barycenter X is stationary at the common center. Over the family, thelocus of X and X are concentric ellipses (solid red and green). The power of the center wrt thecircumcircle (dashed red) and Euler’s circle (dashed green) is invariant. This family conserves area and sum of squared sidelengths, and consequently theBrocard angle [17]. The Barycenter X is stationary at O . The Cayley conditionimplies that ( a c , b c ) = ( a/ , b/ .Over this family, the locus of both X and X are ellipses concentric and axis-aligned with the original pair [8]. Their semi-axes are given by: HELMAN, LAURAIN, GARCIA, AND REZNIK ( a , b ) = c (cid:18) a , b (cid:19) , and ( a , b ) = c (cid:18) a , b (cid:19) Nevertheless:
Proposition 3.
Over 3-periodics in the homothetic pair, the power of the center O = X with respect to both the circumcircle and Euler circle are invariant andgiven by: P = − a + b , P = − a + b Proof.
Squared radii and squared distances between barycenter X and circumcen-ter X or nine-point center X can be obtained via direct computation and in termsof squared sidelengths. This yields: P = − X s i , P = − X s i Recall the sum of squared sidelengths is conserved in the homothetic pair andgiven by [17, Remark 2.1]: X s i = 92 ( a + b ) (cid:3) Confocal Pair.
Consider a confocal pair of ellipses which admits a 3-periodicfamily (elliptic billiard); see Figure 5. Classic invariants include perimeter andJoachmisthal’s constant [21]. A recent result is that the sum of angle cosines isinvariant [18, 11]. The Mittenpunkt X is stationary at O [18]. The semi-axes ofthe inner ellipse are given by [7]:(1) [ a c , b c ] = 1 c (cid:2) a ( δ − b ) , b ( a − δ ) (cid:3) where δ = a − a b + b .Furthermore, over said family, the locus of both X and X are concentric, axis-aligned ellipses with semi-axes are given by [10]: [ a , b ] = (cid:20) a − δ a , δ − b b (cid:21) [ a , b ] = (cid:20) − w ( a, b ) + w ( a, b ) δw ( a, b ) , w ( b, a ) − w ( b, a ) δw ( b, a ) (cid:21) where w ( u, v ) = 4 u ( u − v ) , w ( u, v ) = u ( u + 3 v ) , w ( u, v ) = 3 u + v .Recall that over 3-periodics in the confocal pair, P = − δ [11, Thm. 3]. Weextend this to P : Proposition 4.
Over 3-periodics in the confocal pair, the power of the center O = X with respect to Euler’s circle is invariant and given by: P = δµη ( µ + η − µ + η + 1 where µ = a/a c and η = b/b c . ONCELET CENTER POWER AND LOCI 7
Proof.
Straightforward computation of power of O = X with respect to circumcir-cle and Euler circle gives: P = − ( s + s + s − s ( s + s ) − s ( s + s ) − s ( s + s ) + 6 s s s ) s s s ( s + s + s − s s − s s − s s ) P = κ ( s + s + s − s ( s + s ) − s ( s + s ) − s ( s + s ))( s + s + s − s s − s s − s s ) where κ = ( s + s − s )( s − s + s )( − s + s + s ) .We use the following parametrization for 3-periodics P P P in the confocalpair. Let ρ denote the invariant r/R ratio (inradius/circumradius), s the invariantsemi-perimeter, and c the cosine of the internal angle at P : s = 2(1 − c ) sρ − c + 2 , s = ( ρc − c + ρ + 1 − w ) s (1 + c )( ρ − c + 2) , s = ( ρc − c + ρ + 1 + w ) s (1 + c )( ρ − c + 2) where h = 1 − ρ , and w = (1 − c )( h − ( c − ρ ) ) . The squared semi-axis lengthsare given by: a = 4(1 + h ) s (3 − h )(3 + h ) , b = 4(1 − h ) s (3 + h )(3 − h ) Substituting s , s , s in P and P we get powers with respect to the circles: P = − h ) s (9 − h ) , P = − (3 − h )(1 − h ) s (9 − h ) which doesn’t depend of variable parameter c .These values are always negative since O is interior to both the circumcircle andEuler circle.Finally, using a c /a = (1 + h ) / and b c /b = (1 − h ) / we get the formulas in theproposition. (cid:3) Excentral to Confocals.
Referring to Figure 6, the excentral family com-prises the excentral triangles to 3-periodics in the elliptic billiard. Abusing nota-tion, here we let a, b denote the axes of said elliptic billiard (i.e., the caustic to theexcentral family), and a e , b e denote the axes of outer ellipse E , which in [7] wasderived as:(2) a e = ( a + δ ) /b, b e = ( b + δ ) /a where δ is as in (1). Since the Euler’s circle of an excentral triangle is the circum-circle of the reference: Observation 1.
Over the excentral family P = − δ . Still referring to Figure 6:
Proposition 5.
In the family of excentral triangles to the confocal family, P isinvariant and given by: P = − a − b − δ HELMAN, LAURAIN, GARCIA, AND REZNIK
Figure 5.
In the confocal pair (elliptic billiard), the locus of both X and X are concentricellipses, (red and green), axis aligned with the original ones. The power of the center O = X wrtto either the circumcircle (dashed red) or Euler’s circle (dashed green) is invariant. Proof.
In the billiard family, straightforward computation of power of O = X withrespect to Bevan circle (circumcircle of excentral triangle) gives: P = − s s s ) ( s + s + s − s s − s s − s s ) or P = − s (9 − h ) = − a − b − δ (cid:3) Note that P for the excentral family is equivalent to the power of the power ofthe center with respect to the Bevan circle in the confocal family (circumcircle ofexcenters [24, Bevan Circle]). Summary.
Table 1 summarizes some results in this section. As it will be seen inthe next Section, these are special cases of Theorem 1.In Appendix A we show several additional circle-family combinations over whichthe power of the center is invariant.4.
Concentric and Unaligned
We generalize the previous results by showing that over Poncelet 3-periodics inconcentric, non-axis-aligned ellipse pairs, the power of the center with respect to
ONCELET CENTER POWER AND LOCI 9
Figure 6.
The excentral family (solid blue) has an elliptic billiard as its caustic, and billiard 3-periodics (dashed blue) as orthic triangles. The power of the symmedian point X , stationary atthe center, with respect to circumcircle (dashed red) and Euler circle (dashed green) is invariant.Also shown are the elliptic loci of their centers X and X (solid red, solid green), respectively. family N = 3 Cayley P P incircle r = ( ab ) / ( a + b ) − ab − ab ( a + b ) / (2( a + b ) ) circumc. a c + b c = R − R − a c b c homoth. a c = a/ , b c = b/ − ( a + b ) / − ( a + b ) / confocal see (1) − δ δµη ( µ + η − / ( µ + η + 1) excentral see (2) − a − b − δ − δ Table 1.
Summary of invariant power of origin wrt to circumcircle and Euler’s circle for variousconcentric, axis-aligned systems. Recall µ = a/a c and η = b/b c . either circumcircle or Euler’s circle is still invariant. This holds despite (i) theircenters moving along non-axis aligned ellipses and (ii) their radii being variable.The Cayley condition for the concentric version ( O c = O ) of the pair in Figure 1which admits a 3-periodic family reduces to [5]: a b + cos θ c ( a c − b c ) − ( aa c + bb c ) = 0 Note cos θ can be expressed in terms of a, b, a c , b c : cos θ = ( aa c + bb c ) − a b c ( a c − b c ) As shown ion Figure 7, the feasible region of a c , b c lies between to lines. Notethat when θ = 0 the above reduces to: a c a + b c b = 1 Figure 7.
For a given choice of a and b (in the figure, a = 1 . , b = 1 , the feasible region for a c and b c lies between the limit lines where cos ( θ ) is 0 or 1, respectively. Referring to Figure 8, consider the family of 3-periodics interscribed between aconcentric pair of ellipses E and E c , at an angle θ with each other. Consider thepair of ellipses: E : ( b + c ) x − acxy + a y − b a = 0 E c : x a c + y b c − where − c a c + ( ab + ab c + a c b )( ab − ab c − a c b ) = 0 . Proposition 6.
The locus of X and X are ellipses E , and E which are concen-tric with the pair. Furthermore, E is axis-aligned with E c and its aspect ratio isequal to b c /a c . E is given by: E : 4 b a c (cid:0) a c x + b c y (cid:1) − b c (cid:16)(cid:0) a c − b c + b (cid:1) a − a c b ( a c − b c ) (cid:17) = 0 The axes of E are given by a = √ aa c b c + ab b c − ab c − a c b − a c bb c )8 a c bb = √ aa c − a c bb c + a ( b − b c ))8 ba c Observation 2.
For the special case where the pair is axis-aligned ( θ = 0) theexpression for E is tractable, and given by: E : 16 a x ( a − a a c + ab − a c b ) + 16 b a y ( a a c − ab + 3 a c b ) − Now we are in a position to prove our first main result:
ONCELET CENTER POWER AND LOCI 11
Figure 8.
Over the family of 3-periodics (blue) interscribed in a pair of concentric, unalignedellipses, the locus of X and X are also ellipses (red and green) which are concentric with theoriginal pair. Furthermore, the locus of X is axis-aligned with the inner ellipse. Remarkably, thepower of the common center O with respect to either the circumcircle (dashed red) or Euler’s circle(dashed green) is invariant. Theorem 1.
The power of the common center O is invariant with respect to eitherthe circumcircle or the Euler circle and given by: P = − ab c ba c (cid:0) b + a c − b c (cid:1) − ( a c − b c ) P = − ab c ba c (cid:0) b + a c − b c (cid:1) + b c Proof.
We write explicit parametrized expressions for the vertices of 3-periodicsin a generic concentric pair. We write out explicit expressions for circumcenter,Euler center, and circumradius and finally the power of the center with respect tothese. Via a process of laborious manual CAS-based simplification, we arrive at theresult. (cid:3)
Notice that P − P = a c + b c . Observation 3.
When the ellipses are concentric and axis-aligned, Theorem 1reduces to: P = − a c a c − b , P = − a c ( a − a c ) (cid:0) a + b (cid:1) a Referring to Figure 9, consider the concentric, unaligned pair of ellipses E and E c given by: E : x a + y b − , E c : ( b c + ζ ) x − a c ζxy + a c y − b c a c = 0 where ζ is defined relative to θ as follows: tan 2 θ = 2 aζc − ζ Figure 9.
In the concentric-tilted pair (black ellipses), the loci of X k , k = 2 , , , , are ellipses.The Euler line (dashed brown) connect said centers. Furthermore the Cayley condition reduces to: a ζ − ( ab + ab c + a c b )( ab − ab c − a c b ) = 0 Proposition 7.
Over 3-periodics in the concentric, non-axis aligned ellipse pair,the loci of X and X are ellipses concentric and axis-aligned with the outer one,given by: X : 9 b x + 9 a y + ab (4 a b − ab ) = 0 X : ( a − b ) a − ab ( a + b ) a b + a b − a ( a x + b y ) = 0 Generalizing invariant center power.
Recall a pencil of coaxial circles hascollinear centers and identical limiting points [24, Coaxal Circles].Consider the well-known set of 6 coaxial circles [24, Coaxal System] listed inTable 2.Referring to Figure 10. Numerically:
Observation 4.
The power of the center wrt to all coaxial circles listed on Table 2is invariant, with the exception of the tangential circle.
Conjecture 1 in Section 7 presents an experimental generalization to the above.
Generalizing elliptic loci.
The following is a special case of Theorem 2, in Sec-tion 6:
Proposition 8.
If a triangle center X γ = (1 − γ ) X + γX is a fixed affine com-bination of X and X for some γ ∈ R , its locus over 3-periodics in the concentric,non-axis-aligned ellipse pair will be an ellipse. ONCELET CENTER POWER AND LOCI 13 name center squared radiusCircumcircle X R Euler’s circle X R / Steiner orthoptic circle X P s i / Orthocentroidal X R − P s i / Polar † circle X R − ( P s i ) / Tangential ‡ circle X R / (16 | Q cos θ i | ) Table 2.
Six coaxial circles with centers on the Euler line [24, Coaxal System]. † The squaredradius of the polar circle is positive (resp. negative) for obtuse (resp. acute) triangles. Its signedsquared radius is used when computing circle power. ‡ The tangential circle is the only in the listwhose center is not a fixed linear combination of X and X . Figure 10.
Top:
An acute 3-periodic (blue) is shown in a concentric, unaligned pair. The powerof the center is invariant with respect to the following set of coaxial circles whose centers lie atlinear combinations of X and X : circumcircle (red), Euler’s (green), Steiner’s inellipse orthoptic(brown), orthocentroidal (magenta), centered on X k , k = X is not a fixed linear combination of X and X . Bottom: an obtuse 3-periodic (blue)is shown in a concentric, unaligned pair. All coaxial circles now intersect at two common points.The polar circle (cyan) is now defined, centered on X . The power of the center wrt to all circlesshown (except for the tangential, dashed orange) is constant. There are 226 triangle centers on [13] which are fixed linear combinations of X and X ; see Observation 8. Experimentally, these are also the only ones whichtrace out ellipses.Consider the converse of Proposition 8. If for some concentric, non-axis alignedpair some triangle center X has an elliptic locus, can it be affirmed that X is afixed linear combination of X and X ? Consider the case of the Steiner point X .This point is known to lie on the Euler line, circumcircle, and Steiner circumellipse,although it is not a fixed linear combination of X and X [13]. Still, over (i) thehomothetic family, and (ii) the family with circumcircle, its locus is (i) the outer(Steiner) ellipse, and (ii) the outer circle. However, over the confocal family, thelocus of X is non-elliptic.Based on experimental evidence, in Conjecture 3 (Section 7) we state a converseto Proposition 8. Non-Concentric with Circumcircle
Here we consider 3-periodics inscribed in a circle and circumscribing a non-concentric ellipse. We will work in the complex plane and apply Blaschke Prod-uct techniques [4] which simplify our parametrization. Namely, 3-periodic verticesbecome symmetric with respect to the information of the circle-ellipse pair.As a first step, identify points in R with points in the complex plane C . Let D denote the open unit disk { z ∈ C : | z | < } and T denote the unit circle { z ∈ C : | z | = 1 } . By translation and scaling, we may assume the outer circle ofthe pair to be the unit circle T . Let { f, g } be the two foci of the inner ellipse. Asin [4], define: Definition 2.
Degree-3 Blaschke Product B ( z ) := z (cid:18) z − f − f z (cid:19) (cid:18) z − g − gz (cid:19) Note that if one wants to study the concentric setting, just substitute g = − f .Following chapter 4 of [4], for each λ ∈ T , the three solutions of B ( z ) = λ arethe vertices of a 3-periodic orbit of the Poncelet family of triangles in the complexplane, and as λ varies in T , the whole family of triangles is covered. Clearing thedenominator in this equation and passing everything to the left-hand side, we get z − ( f + g + λf g ) z + ( f g + λ ( f + g )) z − λ = 0 Let z , z , z ∈ C denote the vertices of Poncelet 3-periodics in the pair withwith circumcircle. Using Viète’s formula, we obtain the following parametrizationof the elementary symmetric polynomials on z , z , z : Definition 3 (Blaschke’s Parametrization) . σ := z + z + z = f + g + λf gσ := z z + z z + z z = f g + λ ( f + g ) σ := z z z = λ where f, g are the foci of the inner ellipse and λ ∈ T is the varying parameter.Referring to Figure 11: Proposition 9.
If a triangle center X α,β = αX + βX is a fixed linear combinationof X and X for some α, β ∈ C , its locus over 3-periodics in the non-concentricpair with a circumcircle is a circle centered on O α and of radius R α given by: O α = α ( f + g )3 , R α = | αf g | Observation 5.
Notice that the center and radius of the locus do not depend on β since the circumcenter X is stationary at the origin of this system.Proof. Since, z , z , z are the 3 vertices of the Poncelet triangle inscribed in theunit circle, its barycenter and circumcenter are given by X = ( z + z + z ) / and X = 0 , respectively. We define X α,β := αX + βX = α ( z + z + z ) / .Using Definition 3, we get X α,β = α ( f + g + λf g ) / α ( f + g ) / λ ( αf g ) / ,where the parameter λ varies on the unit circle T . Thus, the locus of X γ over the ONCELET CENTER POWER AND LOCI 15
Poncelet family of triangles is a circle with center O α := α ( f + g ) / and radius R α := | αf g | / | αf g | / . (cid:3) Using α = 1 − γ, β = γ for a fixed γ ∈ R in Proposition 9, we get: Corollary 2.
If a triangle center X γ = (1 − γ ) X + γX is a real affine combinationof X and X for some γ ∈ R , its locus over 3-periodics in the non-concentric pairwith a circumcircle is a circle. Moreover, as we vary γ , the centers of these loci arecollinear with the fixed circumcenter. Many triangle centers in [13] are affine combinations of the barycenter X andcircumcenter X . See Observation 8 for a compilation of them. Observation 6.
For a generic triangle, only X , and X are simultaneously onthe Euler line and on the circumcircle. However these are not linear combinationsof X and X . Still, if a triangle center is always on the circumcircle of a generictriangle (there are many of these, see [24, Circumcircle] ), its locus over 3-periodicsin the non-concentric pair with circumcircle is trivially a circle. Corollary 3.
Over the family of 3-periodics inscribed in a circle and circumscribinga non-concentric inellipse centered at O c , the locus of X k , k in 2,4,5,20 are circleswhose centers are collinear. The locus of X is centered on O c . The centers andradii of these circular loci are given by: O = f + g , O = f + g, O = f + g , O = − ( f + g ) r = | f g | , r = | f g | , r = | f g | , r = | f g | Proof.
As in Corollary 2, we can use Proposition 9 with γ = 0 , − , − / , to getthe center and radius for X , X , X , X , respectively. All of these centers are realmultiples of f + g , so they are all collinear. Moreover, the center O of the circularloci of X is ( f + g ) / , that is, the midpoint of the foci of the inellipse, or in otherwords, the center O c of the inellipse. (cid:3) Referring to Figure 11:
Observation 7.
The family of 3-periodics in the pair with circumcircle includesobtuse triangles if and only if X is exterior to the caustic. This is due to the fact that when X is interior to the caustic, said triangle centercan never be exterior to the 3-periodic. Conversely, if X is exterior, it must alsobe external to some 3-periodic, rendering the latter obtuse.6. Generic Nested Ellipses
In this Section we prove the locus of a given fixed linear combination of X and X is an ellipse. We will continue to use Blaschke product techniques since ageneric non-concentric pair can always be seen as the affine image of a pair withcircumcircle.Consider the generic pair of nested ellipses E = ( O, a, b ) and E c = ( O c , a c , b c .θ ) inFigure 1. Let s θ , c θ denote the sine and cosine of θ , respectively. Define c c = a c − b c .The following is the Cayley condition for the pair to admit a 3-periodic family: Figure 11.
Left: X , i.e., all 3-periodics are acute. The loci of X and X are interior to the circumcircle. Right: X is exterior to the caustic, and 3-periodics can be either acute or obtuse. Equivalently, the locusof X intersects the circumcircle. In both cases (left and right), the loci of X k , k in 2,4,5,20 arecircles with collinear centers (magenta line). The locus of X is centered on O c . The center of the X locus is at / along OO c . b x c + 2 a b x c y c + (cid:0) c c (cid:0) − b ( a + b ) (cid:1) c θ − (cid:0) b − b c (cid:1) b a − b b c (cid:1) x c − a b x c y c c c s θ c θ + a y c + (cid:0) c c a (cid:0) a + b (cid:1) c θ − (cid:0) b c + b (cid:1) a + 2 a b b c (cid:1) y c + c c c (cid:0) c θ − c c c (cid:0) a a c − b a + b c b (cid:1) c θ + ( aa c + ab − bb c ) ( aa c − ab − bb c ) ( aa c + ab + bb c ) ( aa c − ab + bb c ) = 0 Before moving on, we first prove a small parametrization lemma for complexcoordinates:
Lemma 1. If u, v, w ∈ C and λ is a parameter that varies over the unit circle T ⊂ C , then the curve parametrized by F ( λ ) = uλ + v λ + w is an ellipse centered at w , with semiaxis | u | + | v | and (cid:12)(cid:12) | u | − | v | (cid:12)(cid:12) , rotated with respectto the canonical axis of C by an angle of (arg u + arg v ) / .Proof. If either u = 0 or v = 0 , the curve h ( T ) is clearly the translation of a multipleof the unit circle T , and the result follows. Thus, we may assume u = 0 and v = 0 .Choose k ∈ C such that k = u/v . Write k in polar form, as k = rµ , where r > ( r ∈ R ) and | µ | = 1 . We define the following complex-valued functions: R ( z ) := µz, S ( z ) := rz + (1 /r ) z, H ( z ) := kvz, T ( z ) := z + w One can straight-forwardly check that F = T ◦ H ◦ S ◦ R .Since | µ | = 1 , R is a rotation of the plane, thus R sends the unit circle T to itself.Since r ∈ R , r > , if we identify C with R , S can be seen as a linear transformationthat sends ( x, y ) (( r + 1 /r ) x, ( r − /r ) y ) . Thus, S sends T to an axis-aligned,origin-centered ellipse E with semiaxis r + 1 /r and | r − /r | . H is the compositionof a rotation and a homothety. H sends the ellipse E to an origin-centered ellipse ONCELET CENTER POWER AND LOCI 17 E rotated by an angle of arg( kv ) = arg( k )+arg( v ) = (arg( u ) − arg( v )) / v ) =(arg( u ) + arg( v )) / . The semiaxis of E have length | kv | ( r + 1 /r ) = r | v | ( r + 1 /r ) = | r v | + | v | = | k v | + | v | = | u | + | v | , and | kv || r − /r | = r | v || r − /r | = (cid:12)(cid:12) | r v | − | v | (cid:12)(cid:12) = (cid:12)(cid:12) | k v | − | v | (cid:12)(cid:12) = (cid:12)(cid:12) | u | − | v | (cid:12)(cid:12) Finally, T is a translation, thus T sends E to an ellipse E centered at w , rotatedby an angle (arg( u ) + arg( v )) / from the axis, with semiaxis lengths | u | + | v | and (cid:12)(cid:12) | u | − | v | (cid:12)(cid:12) , as desired. (cid:3) Recall that over Poncelet N-periodics interscribed in a generic pair of conics, thelocus of vertex and area centroids is an ellipse [20] as is that of the circumcenter-of-mass [22], a generalization of X for N > . Referring to Figure 12: Theorem 2.
Over the family of 3-periodics interscribed in an ellipse pair in generalposition (non-concentric, non-axis-aligned), if X α,β is a fixed linear combination of X and X , i.e., X α,β = αX + βX for some fixed α, β ∈ C , then its locus is anellipse.Proof. Consider a general N = 3 Poncelet pair of ellipses that forms a 1-parameterfamily of triangles. Without loss of generality, by translation and rotation, we mayassume the outer ellipse is centered at the origin and axis-aligned with the plane R , which we will also identify with the complex plane C . Let a, b be the semi-axisof the outer ellipse, and a c , b c the semi-axis of the inner ellipse, as usual.Consider the linear transformation that takes ( x, y ) ( x/a, y/b ) . This transfor-mation takes the outer ellipse to the unit circle T and the inner ellipse to anotherellipse. Thus, it transforms the general Poncelet N = 3 system into a pair wherethe outer ellipse is the circumcircle, which we can parametrize using Blaschke prod-ucts [4]. In fact, to get back to the original system, we must apply the inversetransformation that takes ( x, y ) ( ax, by ) . As a linear transformation from C to C , we can write it as L ( z ) := pz + qz , where p := ( a + b ) / , q := ( a − b ) / .Let z , z , z ∈ T ⊂ C be the three vertices of the circumcircle family, parametrizedas in Definition 3, and let v := L ( z ) , v := L ( z ) , v := L ( z ) be the three verticesof the original general family. The barycenter X of the original family is given by ( v + v + v ) / , and the circumcenter X is given by [23]: X = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v | v | v | v | v | v | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v v v v v v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Since z = 1 /z , z = 1 /z , z = 1 /z , we can write v , v , v as rational functionsof z , z , z , respectively. Thus, both X and X are symmetric rational functionson z , z , z . Defining X α,β = αX + βX , we have consequently that X α,β is also asymmetric rational function on z , z , z . Hence, we can reduce its numerator anddenominator to functions on the elementary symmetric polynomials on z , z , z .This is exactly what we need in order to use the parametrization by Blaschkeproducts.In fact, we explicitly compute: X α,β = p q (cid:0) σ ( α + 3 β ) + 3 βσ (cid:1) + αp σ σ − pq (3 β + σ σ ( α + 3 β )) − αq σ σ ( p − q )( p + q ) where σ , σ , σ are the elementary symmetric polynomials on z , z , z . Let f, g ∈ C be the foci of the inner ellipse in the circumcircle system. UsingDefinition 3, with the parameter λ varying on the unit circle T , we get: X α,β = uλ + v λ + w , where u := p (cid:0) f g (cid:0) αp − q ( α + 3 β ) (cid:1) + 3 βpq (cid:1) p − q )( p + q ) v := βpq ( q − f gp )( q − p )( p + q ) + 13 αf gqw := q (cid:0) f + g (cid:1) (cid:0) p ( α + 3 β ) − αq (cid:1) + p ( f + g ) (cid:0) αp − q ( α + 3 β ) (cid:1) p − q )( p + q ) By Lemma 1, this is the parametrization of an ellipse centered at w , as desired.As in Lemma 1, it is also possible to explicitly calculate its axis and rotation angle,but these expressions become very long. (cid:3) Corollary 4.
Over the family of 3-periodics interscribed in an ellipse pair in gen-eral position (non-concentric, non-axis-aligned), if X γ is a real affine combinationof X and X , i.e., X γ = (1 − γ ) X + γX for some fixed γ ∈ R , then its locus isan ellipse. Moreover, as we vary γ , the centers of the loci of the X γ are collinear.Proof. Apply Theorem 2 with α = 1 − γ, β = γ to get the elliptical loci. As inthe end of the proof of Theorem 2, the center of the locus of X γ can be computedexplicitly as w = w + w γ , where w = 13 (cid:0) q (cid:0) f + g (cid:1) + p ( f + g ) (cid:1) w = q (cid:0) p + q (cid:1) (cid:0) f + g (cid:1) − p ( f + g ) (cid:0) p + 2 q (cid:1) p − q )( p + q ) As γ ∈ R varies, it is clear the center w sweeps a line. (cid:3) We proved that all of the following triangle centers have elliptic loci in thegeneral N=3 Poncelet system, including the barycenter, circumcenter, orthocenter,nine-point center, and de Longchamps point:
Observation 8.
Amongst the 40k+ centers listed on [13] , about 4.9k triangle cen-ters lie on the Euler line [14] . Out of these, only 226 are fixed affine combinationsof X and X . For k < , these amount to X k , k =
2, 3, 4, 5, 20, 140, 376, 381,382, 546, 547, 548, 549, 550, 631, 632 . Observation 9.
The elliptic loci of X and X are axis-aligned with the outerellipse. We conclude this section with phenomenon specific to the case where E c is acircle, Figure 13: Observation 10.
Over the family of 3-periodics inscribed in an ellipse and cir-cumscribing a non-concentric circle centered on O c = X , the locus of X and X are ellipses whose major axes pass through X . ONCELET CENTER POWER AND LOCI 19
Figure 12.
A 3-periodic is shown interscribed between two nonconcentric, non-aligned ellipses(black). The loci of X k , k = 2 , , , , (and many others) remain ellipses. Those of X and X remain axis-aligned with the outer one. Furthermore the centers of all said elliptic loci are collinear(magenta line). Conjectures and Videos
Experimentally, we can generalize Observation 4 as follows:
Conjecture 1.
Over 3-periodics in the generic concentric pair, the power of com-mon center with respect to some circle C X is invariant if its center X is a fixedlinear combination of X and X and the circle is coaxial with the circumcircle andEuler’s circle. Referring to Figure 14:
Conjecture 2.
Over 3-periodics in the non-concentric, non-axis-aligned pair, thereis some fixed point P (resp. P ) such that its power with respect to the circumcircle(resp. Euler circle) is invariant. In [19, 7] it is shown that in the confocal pair the locus of the incenter X is anellipse.Experimentally, we can strengthen Proposition 8: Figure 13.
A 3-periodic (blue) is shown inscribed in an outer ellipse and an inner non-concentriccircle centered on O c . The loci of both circumcenter (solid red) and Euler center (solid green) areellipses whose major axes pass through O c . Figure 14.
Consider a 3-periodics (blue) in a pair of ellipses in general position (centers at O and O c ), as well as its circumcircle (red, center X ) and Euler’s circle (green, center X ). A point P can be numerically located whose power to the circumcircle (solid red) is invariant over the3-periodic family. Iso-curves of the variance of power of ( x, y ) with respect to the circumcircleare shown (dashed red): the minimum (and zero) variance occurs at P . An analogous numericapproach is used to locate the point P whose power wrt Euler’s circle (solid green) is invariant. Conjecture 3.
If the locus of a triangle center X is an ellipse for all concentric,non-axis-aligned ellipse pairs, then X is a fixed linear combination of X and X . Our very first experimental result was that over 3-periodics in the confocal pair,the locus of the incenter X was an ellipse [16]. This was subsequently proved[19, 7]. Considering the space of choices of nested ellipse pairs is 5d (5 parametersfor each minus 4d homothethy group, minus 1d Cayley condition), little did weknow how rare a phenomenon that was (the space of confocal ellipses is a 1d): ONCELET CENTER POWER AND LOCI 21
Circle Description (see [24]) Center RadiusAnticompl. Circumc. of Anticompl. X R Bevan Circumc. of Excentral X R Spieker Incircle of Medial X r/ Mandart Circumc. of Extouch X see (3) Table 4.
Definition of certain circles for which under certain families the power of the center isalso invariant.
R, r denote circumradius and inradius.
Conjecture 4.
The only pair of ellipses admitting Poncelet 3-periodics such thatthe locus of the incenter X is an ellipse is the confocal pair. Videos.
Animations illustrating some phenomena herein are listed on Table 3. id Title youtu.be/<.>
01 Cayley-Poncelet Phenomena I: Basics virCpDtEvJU
02 Cayley-Poncelet Phenomena II: Intermediate
Table 3.
Videos of some focus-inversive phenomena. The last column is clickable and providesthe YouTube code.
Acknowledgements
We would like to thank A. Akopyan for valuable insights. The third author is fellowof CNPq and coordinator of Project PRONEX/ CNPq/ FAPEG 2017 10 26 7000508.
Appendix A. Additional Invariant-Power Circles
Consider the well-known circles derived from a reference triangle and their radii,listed on Table 4.Let s denote the semiperimeter. The radius of the Mandart circle is given by[24, Mandart Circle]:(3) R m = ss s s p (4 R − s s )(4 R − s s )(4 R − s s ) Referring to Figure 15, for selected families, the power of the center is alsoinvariant.
Proposition 10.
Over the homothetic family, the power of the center with respectto the anticomplementary circle is given by: P ,act = − (4 / X s i Proposition 11.
Over the confocal family, the power of the center with respect tothe Bevan, Spieker, and Mandart circles are given by: P ,bev = − ( a + b + 2 δ ) P ,spi = − (1 / − h ) ( a + b + 2 δ ) P ,man = − (1 / − h + 14 h + 3)( a + b + 2 δ ) Figure 15.
Four additional circles (Anticomplementary, Bevan, Spieker, Mandart) with respectto which the power of the center is invariant, for the particular families indicated (homothetic,and thrice confocal, respectively). See these in motion at: bit.ly/3qiTsgY , bit.ly/3jPVvqf , bit.ly/3bhOVFp , bit.ly/3poGBbW , respectively. Family Triangle Circle AnimationConfocal Extouch Euler’s bit.ly/3phwBkz
Incircle Intouch Euler’s bit.ly/2ZapHD0
Homothetic Medial Euler’s bit.ly/3rTNQdc
Circumcircle Euler’s ‡ Euler’s bit.ly/3qjytdY
Dual † Euler’s Euler’s bit.ly/2Nlx7AS
Dual Anticompl. Circumc. bit.ly/3ai7BWg
Circumcircle Orthic Incircle bit.ly/3qhjhy0
Table 5.
Experimentally, the power of the center wrt certain additional family-circle combinationsis also invariant. ‡ Euler’s Triangle [24] has vertices at the midpoints of lines from the orthocenter X to the vertices (they lie on Euler’s circle). with h = ( − a − b + 2 δ ) /c . Additional circles and families show on Table 5 have been detected experimen-tally, with respect to which the center has constant power. We challenge the readerto derive them.All of the examples in this section can be viewed in motion in bit.ly/37dr1JJ.
ONCELET CENTER POWER AND LOCI 23
Appendix B. Table of Symbols symbol meaning E , E c outer and inner ellipses O, O c centers of E , E c a, b, a c , b c outer and inner ellipse semi-axes’ lengths c, c c half-focal length of E , E c O c = ( x c , y c ) θ major semi-axis tilt E c wrt E P i , s i r, R a i , b i semiaxes of the locus of X i r i radius of the locus of X i (if a i = b i ) C , C circum- and Euler’s circle C , C Steiner orthoptic and orthocentroidal circle C , C polar and tangential circle P i power of center O wrt C i X , X , X Incenter, Barycenter, Circumcenter X , X , X Orthocenter, Euler center, Symmedian point X , X Mittenpunkt, de Longchamps point
Table 6.
Symbols used in the article.
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