Related by Similarity II: Poncelet 3-Periodics in the Homothetic Pair and the Brocard Porism
aa r X i v : . [ m a t h . M G ] S e p RELATED BY SIMILARITY II: PONCELET 3-PERIODICS INTHE HOMOTHETIC PAIR AND THE BROCARD PORISM
DAN REZNIK AND RONALDO GARCIA
Abstract.
Previously we showed the family of 3-periodics in the elliptic bil-liard (confocal pair) is the image under a variable similarity transform of poris-tic triangles (those with non-concentric, fixed incircle and circumcircle). Bothfamilies conserve the ratio of inradius to circumradius and therefore also thesum of cosines. This is consisten with the fact that a similarity preserves angles.Here we study two new Poncelet 3-periodic families also tied to each other viaa variable similarity: (i) a first one interscribed in a pair of concentric, ho-mothetic ellipses, and (ii) a second non-concentric one known as the Brocardporism: fixed circumcircle and Brocard inellipse. The Brocard points of thisfamily are stationary at the foci of the inellipse. A key common invariant is theBrocard angle, and therefore the sum of cotangents. This raises an interestingquestion: given a non-concentric Poncelet family (limited or not to the outerconic being a circle), can a similar doppelgänger always be found interscribedin a concentric, axis-aligned ellipse and/or conic pair?
Keywords
Poncelet, Brocard, Homothetic, Porism, Confocal, Billiard
MSC Introduction
Previously we studied invariants in the so-called Poristic triangle family (fixedincircle and circumcircle) [7] in relation to 3-periodics in the elliptic billiard [9]. Wefound both these families were images of one another under a variable similaritytransform. By definition, poristic triangles conserve inradius r and circumradius R ;we showed billiard triangles conserve r/R [10, 18]. Since r/R = 1 + P cos θ i , bothconserve the sum of cosines.Here we study a new duo of Poncelet 3-periodics families which are also relatedby a similarity and also share a common invariant (the sum of cotangents). Thesearise as follows:(1) The homothetic pair: an external ellipse with semi-axes ( a, b ) , and an inter-nal, concentric, axis-aligned one with semi-axes ( a ′ , b ′ ) = ( a/ , b/ . Theseare known as the Steiner circum- and inellipse, respectively.(2) Brocard Porism: a fixed circumcircle and a caustic known as the BrocardInellipse [22, Brocard Inellipse].While the first one preserves area and its barycenter is stationary (it is an affineimage of the concentric circular pair) the second one conserves the Brocard angle ω ;see Figure 1. This family is remarkable as its Brocard points [22, Brocard Points]are stationary at the inellipse foci. Date : September, 2020.
Figure 1.
For every triangle two Brocard Points exist where sides P i P i +1 concur when rotatedan angle ω about P i toward the triangle’s interior. If vertices are traversed counterclockwise (resp.clockwise) one obtains Ω (resp. Ω ). Main results. we (i) prove the homothetic family conserves ω . We then (ii) de-scribe a 2d parametrization for the family of triangles with fixed Brocard points intermos of the axes of the Brocard inellipse. Finally we show that (iii) the homotheticand Brocard-Poristic families are similar. Since this transform is angle-preserving,it is consistent with the fact that both families conserve Brocard angle. In turn thisimplies both conserve the sum of cotangents, since cot ω = P cot θ i [22, BrocardAngle, Eq. 1]. Related Work.
Properties and relations involving the Brocard points have beenwidely studied [5, 7, 12, 19].Johnson terms equibrocardal families which conserve ω and proves that the pro-jection of a family of equilaterals embedded in a tilted plane onto the horizontalplane is equibrocardal [13, Chapter XVII]. Pamfilos describes equibrocardal isosce-les triangles arising from projections on the Lemoine axis [16].Bradley studies the locus of triangle centers over the Brocard porism in [4] anddefines conics associated with the Brocard points and porism in [3]. A constructionfor the Brocard porism is given in [2, Theorem 4.20, p. 129]. Structure of the Paper.
In Section 2 (resp. Section 3) we describe 3-periodicinvariants associated with the homothetic pair (resp. Brocard porism). In Section 4we prove that both families are images of one another under a variable similaritytransform. Section 5 presents a summary of our findings as well as a few openquestions and a list of videos of some phenomena covered herein.Appendices are included which provide (i) explicit expressions for Brocard porismvertices, (ii) a construction for the complete (2d) family of triangles with fixed Bro-card points, and (iii) explicit expressions for the locus of key triangle centers ineither family.
ELATED BY SIMILARITY II: HOMOTHETIC PAIR AND BROCARD PORISM 3
Figure 2.
Area and sum of squared sidelengths of 3-periodics (blue) in the homothetic Ponceletpair (barycenter X stationary at the origin) are invariant, and therefore so is the Brocard angle ω . Video Homothetic Poncelet
Note: when referring to triangle centers below, we will adopt the X k notationafter [14], e.g., X is the barycenter, X the circumcenter, etc.Consider the family of Poncelet 3-periodics in a homothetic pair , i.e., inscribedin an ellipse ( a, b ) and circumscribed about a concentric, axis-aligned, half-sizedellipse with semi-axes ( a ′ , b ′ ) = ( a/ , b/ . Notice this pair satisfies a condition forthe existence of a 3-periodic Poncelet family in a concentric, axis-aligned ellipsepair, namely [11]:(1) a ′ a + b ′ b = 1 Referring to Figure 2:
Lemma 1.
The family of 3-periodics in the homothetic pair conserves both areaand sum of sidelengths squared.
The proof below was kindly contributed by S. Tabachnikov [21].
Proof.
Area conservation stems from the fact that the family is the affine imageof 3-periodics in a concentric circular Poncelet pair. Invariant sum of squaredsidelengths follows from the fact that the average of the harmonics of degree 1and 2 over the group of rotations of order 3 is zero. Namely, consider a unitvector v ( α ) = (cos α, sin α ) and a matrix A taking concentric circles to homotheticellipses. Then |A v ( α ) | is a trigonometric polynomial of degree 2. Average it over Z by adding π/ and π/ to α . The result is independent of α , as needed. (cid:3) DAN REZNIK AND RONALDO GARCIA
Remark . The invariant area and sum of squared sidelengths of 3-periodics in thehomothetic pair are given by: A = 3 √ ab , X s i = 92 ( a + b ) · Theorem 1. ω given by: cot ω = √ a + b )2 ab . Proof.
A known relation is cot( ω ) = P ( s i ) / (4 A ) [22, Brocard Angle]. Since bothnumerator and denominator are conserved the result follows. (cid:3) Surprisingly, the sum of cotangents is invariant for N -periodics in the concentric,homothetic pair for any N [17].Theorem 1 is also a direct consequence to a beautiful Theorem by Johnson [13,Theorem 487, p. 291], namely that if two equilateral triangles in an oblique planeare projected onto a horizontal one, the projected triangles will have the sameBrocard angle. Johnson’s projection can be regarded as the affine transformationthat takes Poncelet equilaterals interscribed between two concentric circles intothe homothetic pair. Let ϕ denote the angle between the oblique planes and thehorizontal. Johnson derives [13, p. 292]: cot ω = √ (cid:18) cos ϕ + 1cos ϕ (cid:19) We show elsewhere [8] that over the homothetic family, the loci of the Brocardpoints are two ellipses rotated an equal in opposite directions about the x -axis.Furthermore, these ellipses are concentric and similar to the Poncelet pair ellipses.3. Brocard Porism
A remarkable porism is known [3, 4, 19] for a family of triangles circumscribedin a circle and circumscribing the so-called
Brocard inellipse [4]. Remarkably, thefamily is equibrocardal (conserves ω ) (a term from [13]) and their Brocard pointsare stationary at the foci of the inellipse.Given a triangle, the outer conic is the circumcircle. The inner one is the onecentered on X with Brianchon point is X , i.e., the touchpoints are the intersec-tions of the cevians through X (the symmedians) with the sidelines [22, BrocardInellipse].Assume the inellipse is centered at (0 , and its semi-axes are ( a, b ) . Proposition 1.
The circumcenter X , circumradius R , and Brocard angle ω ofthe Brocard porism are given by: (2) X = [0 , − cδ b ] , R = 2 a b , cot ω = δ b where δ = √ a − b . Derivable from the above is a known requirement for the Brocard porism to bepossible [19, Eqs. 15–17]:
ELATED BY SIMILARITY II: HOMOTHETIC PAIR AND BROCARD PORISM 5
Figure 3.
Nine positions of the 3-periodic Poncelet family inscribed in a circle (black) and circum-scribing an ellipse (black). All triangles have the same Brocard angle ω and stationary Brocardpoint Ω and Ω which lie at the foci of the (Brocard) inellipse. Their midpoint X is at the centerof the inellipse. Triangle centers X and X are stationary and are concyclic with the Brocardpoints on the Brocard circle (dashed orange) whose center is X . Remark . R ≥ c Corollary 1. c = R sin ω p − ω This stems from a beautiful relation presented by Shail [19] whereby the distancebetween the Brocard points is given by: | Ω − Ω | = 4 R sin ω (1 − ω ) It be shown that the above reduces to c .Explicit expressions for the Brocard vertices appear in Appendix A. A few con-figurations of the family are depicted in Figure 3. DAN REZNIK AND RONALDO GARCIA
All triangles with fixed Brocard Points.
As seen above, the Brocardporism gives rise to a 1d family of triangles whose Brocard points coincide with thefoci of the caustic, i.e., the Brocard inellipse.Consider the 1-parameter family of origin-centered confocal ellipses E λ : E λ : x a − λ + y b − λ − , λ < b , < b < a. Remark . The circumcenter X ,λ , circumradius R λ , and Brocard angle ω λ for eachfamily implied by a choice of a, b, λ are given by: X ,λ = (cid:20) , − c δ √ b − λ (cid:21) , R λ = 2( a − λ ) √ b − λ , cot ω λ = δ √ b − λ where δ = √ a − b − λ . This stems from Proposition 1: replace a with √ a − λ and b with √ b − λ and obtain expressions for t Theorem 2.
The family of triangles circumscribing E λ and inscribed in a circle ofradius R k centered on X ,λ has fixed Brocard points on the foci [ ± c, of E λ . More-over, varying λ ∈ ( −∞ , b ) covers the entire 2d family of triangles with Brocardspoints on [ ± c, .Proof. Follows from Proposition 1. (cid:3)
An alternative, synthetic method for constructing the same fixed-Brocard 2dfamily was kindly contributed by Peter Moses [15] appears in Appendix B.As a curiosity, Appendix D lists some circles whose centers and radii are invariantover the Brocard poristic family.4.
Related by Similarity: Homothetic and Brocard-Poristic Families
The homothetic and Brocard-poristic Poncelet families turn out to be images ofeach other under a variable similarity transform. The fact that it preserves anglesis consistent with both families being equibrocardal.To prove this we only need to show that either (i) the Brocard inellipse in the ho-mothetic pair or (ii) the Steiner circumellipse in the Brocard porism have invariantaspect ratio. In fact, both must be true.Referring to Figure 4:
Lemma 2.
Over the homothetic pair, the Brocard inellipse has invariant semi-axesratio β given by β = √ a + 10 a b + 3 b ab > Proof.
The semi-axes of the Brocard inellipse of a triangle
ABC are given explicitlyin terms of sidelengths a, b, c in [22, Brocard Inellipse, Eqns. 3,4]. Combining itwith the expression for sin ω in [22, Eqn. 7], obtain: √ Γ = 2 sin ω, Γ = a b + a c + b c . Where ∆ is the triangle area. Applying this result to the vertices of 3-periodicsin the homothetic pair (see Appendix A), the result follows. (cid:3) ELATED BY SIMILARITY II: HOMOTHETIC PAIR AND BROCARD PORISM 7
Figure 4.
The center of the Brocard inellipse (purple) is X . Over the homothetic family, itslocus (dashed purple) is an ellipse concentric and axis-aligned with the pair (see Appendix C).Over the family the axes of the Brocard inellipse are variable, however its aspect ratio is constant.Therefore this family can be regarded as the image of the Brocard-poristic family under a variablesimilarity transform. Video Referring to Figure 5:
Lemma 3.
Over the Brocard porism, the Steiner circumellipse has invariant semi-axes ratio σ given by σ = 8 a − b + 4 √ a − a b + b b Proof.
The Steiner circumellipse S ( t ) is centered in X and passes through thevertices P ( t ) , P ( t ) , P ( t ) (see Appendix A. With a computer-aided algebra system(CAS) we obtain the following implicit equation for it: S ( x, y ) = a ( t ) x + 2 a ( t ) xy + a ( t ) y + a ( t ) x + a ( t ) y + a ( t ) = 0 . The ratio of its semi-axes is obtained from its Hessian matrix H as follows [20]: σ ( t ) = tr ( H ) + p tr ( H ) − H )2 det( H ) , Using a CAS, verify σ is independent of t and equal to the expression in theclaim. (cid:3) Theorem 3.
The 3-periodic family in a homothetic pair is similar to 3-periodicsarising from a 1d family of Brocard porisms. Conversely, the 3-periodic family ina Brocard porism pair is similar to 3-periodics arising in a 1d family of homotheticpairs.
DAN REZNIK AND RONALDO GARCIA
Figure 5.
The Steiner Circumellipse (red) and Inellipse (dashed red) for a 3-periodic in the Bro-card porism (blue) is shown. Both ellipses are by definition centered on X whose locus is a circlecentered on C (see Appendix C. Over the family, the lengths of their axes are variable, howevertheir ratio is invariant. Therefore Brocard-poristic 3-periodics can be regarded as the image undera variable similarity transform of the homothetic family. Note this is consistent with both systemsbeing equibrocardal. Video Proof.
Consider the transformation X ′ = Scale ( k/b ) .Rot ( − θ ) .T ransl ( − X ) .X .where k is a chosen constant, b is the variable minor semi-axis length of the the(moving) Brocard inellipse in the homothetic pair, θ the angle between said minoraxis and the horizontal, and X is the moving center of the inellipse. Clearly, thetransformation will take the moving Brocard inellipse to an origin-centered, uprightone. By Lemma 2, the ratio β of inellipse semi-axes a /b is constant, implyingthe transformed inellipse will have fixed axes ( kβ, k ) . Notice its circumcenter andcircumradius are prescribed by the semi-axes of the caustic (Equation 2). So thehomothetic family will be mapped to a 1-parameter family of Brocard porismswhere the parameter is k . (cid:3) ELATED BY SIMILARITY II: HOMOTHETIC PAIR AND BROCARD PORISM 9
Concentric CommonInvariants Non-Concentric
Confocal (Billiard) a , b , X , P s i r/R P cos θ i Chappple Porism r , R , X , X , . . .a /b Homothetic a , b , X , P s i a , b ω P cot θ i Brocard Porism a , b , X , X , X , X , . . .a /b Table 1.
Left column : invariants held in two concentric systems: confocal and homothetic.
Right column : invariants held by their non-concentric, similar counterparts: Chapple- andBrocard-poristic.
Middle column : invariants which hold on both the concentric and non-concentric counterpart. The confocal-Chapple similarity was studied in [9].
The reverse argument can be used to transform the moving Steiner circumellipsein the Brocard porism to an origin centered, stationary one with axes ( k ′ σ, k ′ ) ,namely: X = Scale ( k ′ /b ′ ) .Rot ( − θ ′ ) .T ransl ( − X ′ ) .X ′ Where primed (unprimed) quantities refer to those measured in the Brocardporism (resp. homothetic pair). See Figures 4 and 5 and the videos mentioned inthe captions. 5.
Conclusion
Previously we had shown the similar, concentric doppelgänger to the poristictriangle family (fixed incircle and circumrcircle) were 3-periodics in the confocalpair (elliptic billiard) [9], and that both conserved the sum of cosines. Later weproved the sum of cosines is conserved by the confocal pair for all N [18, 1].Here we make a similar argument: the concentric doppelgänger to 3-periodics inthe non-concentric Brocard porism is the concentric, axis, aligned homothetic family.Both conserve the Brocard angle and therefore the sum of cotangents. Suprisingly,homothetic N-periodics conserve the sum of cotangents for all N [17].These considerations are summarized on Table 1.The following are questions: • are there other (non-concentric,concentric) duos related by similarity? Whatcommon quantity do they conserve? • must the outer conic of the non-concentric pair be a circle so a concentric,similar doppelgänger can be found? • How does this similar duality relate to transformations mentioned in [6]which, under a suitable ambient ( CP ) a transformation exists which takesany non-concentric, non-axis-aligned conic pair into a confocal, canonicalone?A table of animations of some of the results above appears below. Exp Video Title01 * Family of 3-Periodics in Five Poncelet Pairs02 * Poncelet 3-periodics in the Homothetic Pairconserve Brocard angle03 * Brocard-Poncelet Porism with stationaryBrocard Points and invariant Brocard Angle04 * Brocard Porism and Invariant Aspect RatioSteiner Circumellipse05 * Homothetic Pair and Invariant Aspect RatioBrocard Inellipse
Table 2.
Experimental animations. Click on the * to see it as a YouTube video and/or a browser-based simulation.
We would like to thank A. Akopyan who originally challenged us to find an equibro-cardal Poncelet pair. S. Tabachnikov contributed the proof for invariant squaredsidelengths on the homothetic pair applicable to all N . P. Moses contributed sev-eral insights, as well as the geometric construction for the family with fixed Brocardpoints as well as suggesting relevant references. L. Gheorghe pointed us to a rele-vant set of porism results by Bradley and provided valuable feedback on our videos.M. Helman has kindly assisted us with simulations and several insights. The firstauthor is fellow of CNPq and coordinator of Project PRONEX/ CNPq/ FAPEG2017 10 26 7000 508. Appendix A. Brocard Porism: Vertices of 3-periodics
Let the triangle vertices be P i ( t ) , i = 1 , , . Parametrize P ( t ) = X + R [cos t, sin( t )] . ELATED BY SIMILARITY II: HOMOTHETIC PAIR AND BROCARD PORISM 11 P ( t ) =[ 2 a b cos t, − cδ b + 2 a sin tb ] P ( t ) =[ p ,x q , p ,y q ] P ( t ) =[ p ,x q , p ,y q ] p ,x = − a bc (cid:0) a − a b + b (cid:1) cos t − a bc δ sin t cos t − a cδ (cid:0) a − a b + b (cid:1) cos t − a c (cid:0) a − b (cid:1) sin t cos t + a b (cid:0) a − a b + 2 b (cid:1) cos t + 2 a b c δ sin t cos t + a ∆ b (cid:0) a − b (cid:1) sin t + a b cδ ∆ p ,y =8 a bc δ cos t + 4 ∆ a b c cos t − c δ b c (cid:0) a − b (cid:1) cos t − b a c (cid:0) a − b (cid:1) sin t cos t − b ∆ a (cid:0) a − a b + 2 b (cid:1) cos t − b ∆ a cδ (cid:0) a − b (cid:1) sin t cos t − sin ( t ) a b p ,x = − a bc (cid:0) a − a b + b (cid:1) cos t − a bc δ sin t cos t − a cδ (cid:0) a − a b + b (cid:1) cos t − a c (cid:0) a − b (cid:1) sin t cos t + a b (cid:0) a − a b + 2 b (cid:1) cos t + 2 a b c δ sin t cos t + a ∆ b (cid:0) a − b (cid:1) sin t + ∆ a b cδ p ,y =8 a bc δ cos t + 4 ∆ a b c cos t − c δ b (cid:0) a − b (cid:1) cos t − a b c (cid:0) a − b (cid:1) sin t cos t − a b ∆ (cid:0) a cδ − b cδ (cid:1) sin t cos t − a b ∆ (cid:0) a − a b + 2 b (cid:1) cos t − sin ta b q =16 a c cos t − b c (cid:0) a − b δ (cid:1) cos t + a b Appendix B. Peter Moses’ Construction for Fixed Brocard Families
The following construction for the family of triangles with fixed Brocard Pointswas kindly contributed by Peter Moses [15].Given fixed Brocard Points Ω and Ω and vertex A of triangle ABC , compute
B, C as follows: • O a is the circumcenter of A Ω Ω . • A ′ is the reflection of A in the midpoint of Ω Ω . • The perpendicular through A ′ to AO a intersects A Ω at P , A Ω at Q and Ω Ω at R . • A ′′ is the reflection of A in the midpoint of Q Ω . • A ′′′ is the reflection of A in the midpoint of P Ω . • Intersect the circle through A ′′ centered at Q with the conic A Ω RQA ′′ at A ′′ and C . • Intersect the circle through A ′′′ centered at P with the conic A Ω RP A ′′′ at A ′′′ and B .Claim: ABC has Brocard points Ω and Ω . Appendix C. Loci of Homothetic X(39) Brocard-poristic X(2)
C.1.
Homothetic pair: elliptic locus of X(39).
Recall the center of the Bro-card inellipse is X [22, Brocard Inellipse]. Proposition 2.
Over the homothetic pair, the locus of X is an ellipse concentricand axis-aligned with the pair with semi-axes: ( a , b ) = c (cid:18) aa + 3 b , b a + b (cid:19) Proof.
Consider the triangular orbit of the homothetic pair A = [ a cos t, b sin t ] , B = [ a cos( t + 2 π , b sin( t + 2 π , C = [ a cos( t + 4 π , b sin( t + 4 π . Using the trilinear coordinates of X given by [ p : q : r ] = [ a ( b + c ) : b ( a + c ) : c ( a + b )] it follows that X ( t ) = paA + qbB + rcCap + bq + cr , a = | B − C | , b = | A − C | , c = | A − B | . Therefore, using symbolic calculations, it follows that X ( t ) = (cid:20) − a ( a − b ) cos(3 t )2( a + 3 b ) , − b ( a − b ) sin(3 t )2(3 a + b ) (cid:21) · This ends the proof. (cid:3)
C.2.
Brocard porism: circular locus of X(2).
Recall the center of the Steinercircumellipse is X [Steiner Circumellipse][22]. The following was contributed byPeter Moses: Proposition 3.
Over the Brocard porism, the locus of X is a circle centered ontriangle center X and of radius R (2 cos(2 ω ) − / . Note: Trilinear coordinates for X are A − cos( A + 2 ω ) :: [14, Part 6].In addition, the following expressions for the X locus center and radius havebeen derived in terms of a and b : C ( x, y ) = x + y + 2 √ a − a b + b b y + 13 ( a − b ) = 0 centered at [0 , − √ a − a b + b b ] and radius a − b ) / (3 b ) .The circular locus of X can also be inferred from results in [4].We leave it as an exercise: Remark . The axes of the Steiner circumellipse and the circular locus of X inter-sect the minor axis of the Brocard inellipse on the same locations. Appendix D. Stationary Circles over the Brocard Porism
Table 3 lists circles named in [22] which over the Brocard porism have stationarycenters (indicated) and invariant radii (since they only depend on R and ω ).The coordinates for X and radius R of the Brocard circle in terms of a, b are given by: X = [0 , − c √ a − b b √ a − b ] , R = 2 a cb √ a − b · ELATED BY SIMILARITY II: HOMOTHETIC PAIR AND BROCARD PORISM 13
Circle Center RadiusCircumcircle X R X eR Stammler X R † ) X R tan ω Brocard X e ( R/
2) cos ω Gallatly X R sin ω Half-Moses X R sin ω Moses X R sin ω X ( R/
2) sec ω Lucas Inner X R/ (4 cot ω + 7) Table 3.
Circles named in [22] which remain stationary over the Brocard porism. Let e = p − ω . † The Cosine Circle to the excentrals of 3-periodics in the elliptic billiard (confocalpair) is also stationary [10].
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