An Infinite, Converging, Sequence of Brocard Porisms
aa r X i v : . [ m a t h . M G ] O c t AN INFINITE, CONVERGING, SEQUENCE OFBROCARD PORISMS
DAN REZNIK AND RONALDO GARCIA
Abstract.
The Brocard porism is a 1d family of triangles inscribed in a circleand circumscribed about an ellipse. Remarkably, the Brocard angle is invariantand the Brocard points are stationary at the foci of the ellipse. In this paperwe show that a certain derived triangle spawns off a second, smaller, Brocardporism so that repeating this calculation yields an infinite, converging sequenceof porisms. We also show that this sequence is embedded in a continuous familyof porisms.
Keywords
Poncelet, Brocard, Porism, Locus, Invariant, Envelope, Recur-rence, Nesting, Convergence
MSC Introduction
The Brocard points Ω , Ω , shown in Figure 1 are well-studied, unique points ofconcurrence in a triangle, introduced by August L. Crelle in 1816, given a construc-tion by Karl F.A. Jacobi in 1825, and rediscovered by Henri Brocard in 1875 [19,Brocard Points].The Brocard porism , Figure 1, is a 1d family of Poncelet 3-periodics (triangles)inscribed in a circle and circumscribed about an ellipse E known as the Brocard
Date : September, 2020.
Figure 1.
The Brocard Points Ω (resp. Ω ) are where sides of a triangle concur when rotatedabout each vertex by the Brocard angle ω . When sides are traversed and rotated clockwise (resp.counterclockwise), one obtains Ω (resp. Ω ). 1 DAN REZNIK AND RONALDO GARCIA Figure 2.
A 3-periodic (blue)
ABC in the Brocard Porism is shown inscribed in an external circle Γ (black) and the Brocard inellipse E (black). The tangency points are given by intersections D, E, F of symmedians (cevians through X ) with the sidelengths. The Brocard points Ω , Ω are stationary at the foci of E . The Brocard circle K (green) contains Ω , Ω , the circumcenter X and the symmedian point X , all of which are stationary. The center of K is X , themidpoint of X X , aka the Brocard axis. Also shown (purple) is the second Brocard triangle T ′ = A ′ B ′ C ′ inscribed in K whose vertices lie at the intersections of the symmedians with K . Thefirst isodynamic point X is on the Brocard axis, is stationary, and common for both the originaland the T ′ family. Video Inellipse [4]. Remarkably, the Brocard angle ω is invariant and the Brocard points Ω and Ω are stationary at the foci of E .As a review, the circle K which contains both Brocard points and the circumcen-ter X is known as the Brocard circle [19]. The symmedian point X also lies on K and X X is known as the Brocard axis . Notice that over the porism, all saidpoints are stationary, and therefore so is K .At least seven Brocard triangles are known (named 1st, 2nd, etc.) [7], derivingdirectly from Ω , , K and related objects. It turns out the 1st, 2nd, 5th, and 7thBrocard triangles are inscribed in K ; see this Video.Here we focus on the second Brocard triangle, denoted T ′ , whose vertices lie atthe intersections of symmedians (cevians through the symmedian point X ) withthe Brocard circle [19, Second Brocard Triangle]; see Figure 2.One first intriguing observation (proved below), unique to T ′ , is that over theporism, its Brocard points Ω ′ and Ω ′ of T ′ are also stationary; see Figure 3 andthis Video. In turn, this leads to a stream of interesting properties. Main results. • The T ′ are 3-periodics of a second Brocard porism inscribed in K andcircumscribed about a smaller, less eccentric ellipse E ′ . • Recursive calculation of T ′ spawns an infinite sequence of ever-shrinkingporisms which converge to the first isodynamic point X [19, Isodynamic The 7th Brocard triangle was proposed during the course of this research.
N INFINITE, CONVERGING, SEQUENCE OF BROCARD PORISMS 3
Points], common to all families. Successive Brocard points lie on two circu-lar arcs. • Recursive calculation of the inverse T ∗ ) con-verges to a segment between two fixed points P , U known as the Beltramipoints [11]. • Sequential Brocard circles K , K ′ , K ′′ , ... are nested within each other like aRussian doll. • The discrete sequence of porisms is embedded in a continuous family where T ′ (and its inverse) can be regarded as operators which induce discretejumps. • The envelope of ellipses in the continuous porism is an ellipse with theisodynamic points as foci.
Related Work.
A construction for the Brocard porism is given in [2, Theorem4.20, p. 129]. Equibrocardal (Brocard angle conserving) triangle families are stud-ied in [9]. Bradley defines several conics associated to the Brocard porism [3].Odehnal has studied loci of triangle centers for the poristic triangle family [14]identifying dozens of stationary triangle centers; similarly, Pamfilos proves proper-ties of the family of triangles with fixed 9-point and circumcircle [15]. We havepreviously studied triangle center loci over selected triangle families [6, 17]. Wehave also analyzed the Brocard porism alongside the Poncelet homothetic familyestablishing a similarity link between the two [16].
Article Structure. in Section 2 we review definitions and prove basic Brocardrelations required in further sections. In Section 3 we prove properties of the familyof second Brocard triangles in a Brocard porism. In Section 3 we study the infinite,discrete sequence of porisms induced by iterative calculations of the second Brocardtriangle. Finally, in Section 5 we specify how said sequence is embedded in acontinuous one.Appendix A provides explicit expressions for the vertices of an isoceles 3-periodicin the porism. Appendix B provides a 1d parametrization for any 3-periodic in theporism. Appendix C contains additional supporting relations required by some ofour proofs. Finally, Appendix D tabulates most symbols used herein.2.
Properties of Brocard porism triangles
We adopt Kimberling’s X k notation for triangle centers, e.g., X for the incenter, X for the barycenter, etc. [10]. Assume that X = (0 , is at the origin. Referringto Figure 1 Definition 1 (Brocard Points) . Let a reference triangle have vertices
ABC whentraversed counterclockwise. The first Brocard point Ω is where sides AB , BC , CA concur when rotated a special angle ω about A , B , C , respectively. The secondBrocard point Ω is defined similarly by a − ω rotation CB , BA , AC about C , B , A , respectively.Referring to Figure 2: Definition 2 (Brocard Circle) . This is the circle K through X , and the Brocardpoints Ω , Ω . DAN REZNIK AND RONALDO GARCIA
Figure 3.
By construction (dashed purple), the second Brocard triangle T ′ (purple), is inscribed inthe Brocard circle K (green) of the its reference triangle T (blue). Remarkably, over the Brocardporism (outer circle, inner ellipse, both in solid black), the Brocard points Ω ′ , of T ′ are alsostationary. These coincide with the foci of a new, smaller, rounder, Brocard inellipse E ′ (dashedblack). The new, stationary, Brocard circle K ′ (dashed green) is properly contained within K .Notice the upper vertex of E ′ coincides with the Brocard midpoint X of T . The isodynamicpoints X , X (latter not shown, above the page) are common and stationary for T and T ′ .Video Note K also contains X , and it is centered on the midpoint X of X X [19].Referring to Figure 2: Definition 3 (Brocard Porism) . This is a 1d family of Poncelet 3-periodics in-scribed in a circle of radius R (circumcenter X is stationary) and circumscribedabout an ellipse of semi-axes ( a, b ) known as the Brocard inellipse E .Over the family, the Brocard angle ω is invariant and the two Brocard points Ω and Ω are stationary at the foci of E [4]. Corollary 1.
Over the Brocard porism the Brocard circle and X are stationary. This stems from the fact that over the porism Ω , Ω , X are stationary. Since X is antipodal to X on the so-called Brocard axis [19], it is also stationary.After [10]: Definition 4 (Barycentric Combo) . Let P and U be finite points on a triangle’splane with normalized barycentrics ( p, q, r ) and ( u, v, w ) , respectively. Let f and g be homogeneous functions of the sidelengths. The ( f, g ) combo of P and U , alsodenoted f ∗ P + g ∗ U , is the point with barycentrics ( f p + g u, f q + g v, f r + g w ) .The Cyrillic letter ш shall henceforth denote cot ω . Since ω ∈ [0 , π/ [19, BrocardAngle], then: Remark . ш ≥√ . N INFINITE, CONVERGING, SEQUENCE OF BROCARD PORISMS 5
Lemma 1.
Over the porism, the two Isodynamic points X and X are stationaryand given by: X = " , R ( √ − ш ) √ ш − , X = " , − R ( √ ш ) √ ш − Proof.
Peter Moses (cited in [10, X(15), X(16)]) derives the following combos forthe two isodynamic points: X = √ ∗ X + ш ∗ X (1) X = √ ∗ X − ш ∗ X With all involved quantities invariant, the result follows. Note that isodynamicpoints are self-inverses with respect to the circumcircle [19, Isodynamic Points]. Forhow we obtained the explicit expressions see Appendix C. (cid:3)
Lemma 2.
Let R and ω denote a triangle’s circumradius and Brocard angle. Thesemi-axes ( a, b ) and center X of the Brocard inellipse E are given by: [ a, b ] = R (cid:2) sin ω, ω (cid:3) = R (cid:20) √ ш ,
21 + ш (cid:21) X = " , − R ш √ ш − ш + 1 Proof.
Consider a triangle T with sidelengths s , s , s , area ∆ , and circumradius R . The following identities appear in [4, 18]: R = s s s , sin ω = 2∆ √ λ , | Ω − Ω | = 4 c = 4 R sin ω (1 − ω ) where λ = ( s s ) + ( s s ) + ( s s ) (named Γ in [18, Eqn. 2]), and c = a − b .The result follows from combining the above into the following expressions for theBrocard inellipse axes [19, Brocard Inellipse]: a = s s s √ λ , b = 2 s s s ∆ λ . For how explicit expressions were obtained for X , see Appendix C. (cid:3) Proposition 1.
The circumradius R and ш are given by: R = 2 a b , ш = √ a − b b . Proof.
Follows directly from Lemma 2. (cid:3)
DAN REZNIK AND RONALDO GARCIA
Lemma 3.
The coordinates for the symmedian point X are given by: X = " , − R √ ш − ш A derivation is provided in Appendix C.
Corollary 2.
The distance between circumcenter and symmedian point is given by | X − X | = R √ ш − ш Proposition 2.
In terms of R and ш, the Brocard points are given by: Ω , ( R, ш ) = R √ ш − ш + 1 [ ± , − ш ] Proof.
Follows from Proposition 11 and Lemma 9. (cid:3)
Corollary 3.
The center X of the Brocard inellipse E is given by: X = " , − R ш √ ш − ш + 1 Properties of the family of second Brocard triangles
Upwards of seven Brocard triangles are defined in [7]. The 1st, 2nd, 5th, and7th Brocard triangles are inscribed in the Brocard circle, as shown on this video.Henceforth we shall focus on the second Brocard triangle, denoted T ′ = A ′ B ′ C ′ .All primed quantities ( Ω ′ i , ω ′ , etc.) below refer to those of T ′ . Specifically, X ′ i stands for triangle center X i of T ′ .Referring to Figure 2: Definition 5 (Second Brocard Triangle) . The vertices A ′ , B ′ , C ′ of the secondBrocard triangle T ′ lie at the intersections of cevians through X with the Brocardcircle K , i.e., T ′ is inscribed in K .Over the Brocard porism: Lemma 4. X ′ (equivalent to X ) is stationary and given by: X ′ = X = " , − R √ ш − ш This stems from the fact that T ′ is inscribed in fixed K whose center is X .The explicit formula is obtained by noting that X is the midpoint of X X , with X = [0 , and X as given in Lemma 3. Lemma 5. X ′ (equivalent to X ) is stationary and given by: X ′ = X = " , − R ш √ ш − ш + 3 N INFINITE, CONVERGING, SEQUENCE OF BROCARD PORISMS 7
Proof. X ′ is X of the reference triangle [10, X(6)]. The latter is the inverse of X with respect to the Brocard circle [10, X(574)]. In turn, X is the inverse of X with respect to the circumcircle. X is stationary in the porism [4]. Carrying outthe inversions in reverse order, and noting that both the circumcircle and Brocardcircle are stationary, obtain the claim. See Appendix C for a method to obtain theexpression for X . (cid:3) Corollary 4.
The Brocard Circle K ′ of T ′ is stationary. This stems from the fact that stationary X ′ and X ′ are antipodes on K ′ [19,Brocard Circle]. Corollary 5. X ′ and X ′ are stationary.Proof. X ′ (resp. X ′ ) coincide with X (resp. X ) [10, X(15) and X(16)], shownin Lemma 1 to be stationary. (cid:3) Corollary 6.
The T ′ family is equibrocardal, i.e., ω ′ is invariant.Proof. Plugging invariant X ′ , X ′ , X ′ are stationary in the “combo” (1) of Lemma 5yields a unique ш ′ . (cid:3) Let R denote the circumradius of a triangle. After [19, Second Brocard Circle]: Definition 6 (Second Brocard Circle) . Let K denote the circle centered on X through both Brocard points Ω , Ω and with radius R = R p − ω . Lemma 6.
The second Brocard circle K ′ of the T ′ family is stationary.Proof. The T ′ family is inscribed in a fixed circle (i.e., X ′ is stationary and R ′ isinvariant). Since ω ′ is invariant (Corollary 6), so is R ′ , and the result follows. (cid:3) Corollary 7.
The Brocard midpoint X ′ of the T ′ is stationary.Proof. X is the inverse of X with respect to K [10, X(39)]. Since both X ′ and K ′ are stationary (Lemmas 5 and 6), the result follows. (cid:3) Corollary 8. Ω ′ , Ω ′ are stationary.Proof. These both lie on K ′ and their join is perpendicular to the Brocard axis X ′ X ′ , intersecting it at X ′ [19, Brocard Circle]. Since all stationary, the resultfollows. (cid:3) The following results are used to entail Theorem 2.
Lemma 7. X ′ (equivalent to X ) is interior to the segment X ′ X (equivalent to X X ).Proof. Assume X is at the origin. An expression for X was given in Lemma 3and one for X ′ in Lemma 5. Noting that X = X and ш ≥ (Remark 1)yields the result. (cid:3) Proposition 3.
The Brocard circle K ′ of T ′ is contained within its circumcircle,i.e., the Brocard circle K of its reference triangle.Proof. Since X ′ X ′ is a diameter of K ′ , and both are contained within K (seeLemma 5), the result follows. (cid:3) Referring to Figure 3:
DAN REZNIK AND RONALDO GARCIA
Theorem 1.
The family of second Brocard triangles are 3-periodics in a new Bro-card porism specified by: R ′ = R √ ш − ш X ′ = [0 , − R ′ ] ш ′ = ш + 32 ш ( a ′ , b ′ ) = R ′ (cid:18) √ ш ′ + 1 , ш ′ + 1 (cid:19) Ω ′ =Ω ( R ′ , ш ′ ) + X ′ Ω ′ =Ω ( R ′ , ш ′ ) + X ′ X ′ = " , − R ′ (cid:0) ш + 1 (cid:1) ш ′ ш where ш ′ = cot ω ′ and Ω i ( R, ш ) , i = 1 , are as in Proposition 2.Proof. In a general triangle (see Figure 2), the major (resp. minor) axes of theBrocard Inellipse are oriented along Ω Ω (resp. the Brocard axis X X ) and itscenter is the Brocard midpoint X [19, Brocard Inellipse] . Since the Brocardpoints Ω ′ , Ω ′ of T ′ are stationary (Corollary 8), the center of E ′ is stationarycenter. Since X and X are stationary antipodes of K , E ′ is axis-aligned with E .Plug invariant R and ω ′ into the equations in Lemma 2 and obtain invariant ( a ′ , b ′ ) as in the claim. (cid:3) Peter Moses let us know that X ′ is none other than X [13]. Remark . The upper vertex of the inellipse of Brocard E ′ is at the Brocard mid-point X = X ′ + [0 , b ′ ] . Corollary 9.
The semi-axes of E ′ can be expressed in terms of those of E as follows: [ a ′ , b ′ ] = " a √ a − b √ a + 2 b , b √ a − b √ a − b a + 2 b Proof.
This follows from Theorem 1 and Lemma 2. (cid:3) An infinite sequence of porisms
Referring to Figure 4:
Theorem 2.
Recursive calculation of the second Brocard triangle produces an infi-nite sequence of Brocard porisms B ′ , B ′′ , B ′′′ , ... such that: • The isodynamic points are stationary at the original X and X . • The Brocard circle of each new porism is contained within the Brocard circleof its parent, forming an infinite nesting. • Both the circumradius R and the eccentricity ε of the inellipse decreasemonotonically. • The Brocard angle ω increases monotonically and converges to π/ (i.e.,triangles approach equilaterals). • The sequence of porisms converges to X . N INFINITE, CONVERGING, SEQUENCE OF BROCARD PORISMS 9
Proof.
Corollary 5 entails fixed isodynamic points. Proposition 3 entails Brocardcircle infinite nesting. The other results stem from recursive application of relationsin Theorem 1. (cid:3)
Defined in [12, p. 78], and cited in [11, P(2)]:
Definition 7 (Beltrami Points) . Denoted P and U , these are the inverses withrespect to the circumcircle of the Brocard points. They lie on the Beltrami (orLemoine) axis L , parallel to a line through the Brocard points [10, CentralLines].Introduced in [8]: Definition 8. (Anti second Brocard triangle) Given a triangle T , its anti secondBrocard triangle T ∗ is a triangle whose second Brocard is T . Proposition 4.
Recursive calculation of the anti second Brocard triangle producesan infinite sequence of Brocard porisms B ∗ , B ∗∗ , B ∗∗∗ , ... such that: • The isodynamic points remain stationary at the original X and X . • The Brocard circle of each new porism is exterior to the previous one, form-ing a reverse infinite nesting. • The sequence of porisms converges to segment P U , i.e: – The inellipse major (resp. minor) semi-axis converges to | P U | (resp.0). – The circumradius R monotonically increases and converges to infinity. – The Brocard angle ω decreases monotonically to zero.Proof. The result follows applying Theorem 1 and Propositions 12 and 14 observingthat we can invert the process of recurrence taking the sequence of second antiBrocard triangles T ∗ , T ∗∗ , . . . , as isosceles triangles tangent to the Brocard innelipseas shown in Fig. 5. (cid:3) Definition 9 (Beltrami Circles) . Given a triangle, let C (resp. C ) denote thefirst (resp. second) Beltrami circle, passing through the first P (resp. second U )Beltrami point, containing their circumcircle inverse Ω (resp. Ω ). Lemma 8.
The the Beltrami circles C and C are centered on P , U = (cid:20) ∓ R √ ш − , − R ш √ ш − (cid:21) Proof.
The circumcircle inverse of Ω is O = P and that of Ω is O = U .Therefere, P = R Ω | Ω | , U = R Ω | Ω | Therefore the result follows from Proposition 2. (cid:3)
Proposition 5.
The Beltrami circles intersect at X and X and their radii ρ isequal and given by: ρ = 2 R √ ш − Moreover, the triangles X P U and X P U are equilateral. Proof.
Follows from Lemmas 1 and 8. (cid:3)
Theorem 3.
The sequence Ω , Ω ′ , Ω ′′ , Ω ′′′ , etc. (resp. Ω , Ω ′ , Ω ′′′ , Ω ′′′ ) isconcyclic on the first (resp. second) Beltrami circle.Proof. Using Proposition 12, Lemma 9 and Theorem 1 we compute explicitly thesequence of Brocrard points stated. It is straightforward to derive the equation ofthe circles. They are given by C : 4 h (3 d − h )( x + y ) − dh ζx − h (3 d + h ) ζy + (3 d − h ) ζ = 0 C : 4 h (3 d − h )( x + y ) + 8 dh ζx − h (3 d + h ) ζy + (3 d − h ) ζ = 0 where ζ = d + h . The centers are O , = (cid:20) ± dζ d − h , (3 d + h ) ζ h (3 d − h ) (cid:21) = (cid:20) ± R √ ш − , − R ш √ ш − (cid:21) The common radius ρ is given by ρ = 2 dζ d − h = 2 R √ ш − The intersections of C and C are triangle centers X , X of Lemma 1. (cid:3) Using the definition in [19, Isodynamic Points]:
Definition 10. (Apollonius Circles) Given a triangle, there are three circles passingthrough one vertex and both isodynamic points X and X .Let T be the upright isosceles 3-periodic with half base d and height h . Corollary 10.
The Beltrami circles are the first and second Apollonius circles of T . The third Apollonius circle is degenerate and coincides with the Brocard axis.Proof. The anti-second Brocard T ∗ of T has Brocard points Ω ∗ , Ω ∗ which coincidewith vertices B and A of T . Applying Theorem 3 in reverse direction, Ω ∗ , Ω ∗ willlie each on C and C , respectively. Therefore C (resp. C ) passes through vertex B (resp. A ) and the two stationary isodynamic points X , X (anti-Brocardspreserve these). By definition these are the first and second Apollonius circles [19,Isodynamic Points]. The third Apollonius circle contains the two isodynamic pointsand vertex C of T . Since these are collinear on the Brocard axis, the circle is aline. (cid:3) Proposition 6. C and C are perpendicular to each Brocard circle in the sequence.Proof. It is straightforward to verify the claim for the circumcircle of isoscelestriangle T , given implicitly by x + y − R = 0 , R = d + h h . (cid:3) N INFINITE, CONVERGING, SEQUENCE OF BROCARD PORISMS 11
Figure 4.
Three iterations of second Brocard triangles (dashed blue), each spawning its ownBrocard porism. Successive Brocard points descend in alternate fashion along two circular arcs(red). These intersect at X and X (above page, not shown), with centers on the Beltrami points P , U . The sequence of Brocard points and porisms converges to the first isodynamic point X ,common to all porisms. At every generation the circumcircle-inellipse pair approaches a shrinkingpair of concentric circles (the triangle family approaches equilaterals). Video Figure 5.
Sequence of porisms with successive Brocard points walking along two circular arcs (red)bounded by the Beltrami points P , U and X . For each generation the isosceles T A, B , or A ′ , B ′ , etc., of a given T are the Brocardpoints of the previous generation. And that their midpoint coincides with the top vertex of theBrocard inellipse.2 DAN REZNIK AND RONALDO GARCIA Embedding the discrete sequence of porisms in a continuous family
Consider a family of axis-aligned ellipses E t , ≤ t ≤ π/ centered at O t = [0 , − sin t ] with semi-axes a, b given by: a = √ t − , b = p (2 cos t − − cos t ) √ Referring to Figure 6(left):
Remark . The eccentricity of E t is given by ε t = √ t − . The foci f ,t and f ,t lie each on distinct unit-radius circulars arc centered on C , C = [0 , ∓ / ,respectively. Namely: f ,t , f ,t = (cid:20) cos t ± , − sin t (cid:21) Theorem 4 (Continuous) . There is a continuous family of Brocard porisms B t =(Γ t , E t ) whose 1d family of triangles: • is inscribed in circle Γ t centered on X ,t with radius R t given by: X ,t = (cid:20) , sin t t − (cid:21) , R t = s t − − cos t ) • circumscribes E t , its Brocard inellipse (i.e., its Brocard points Ω ,t , Ω ,t are f ,t , f ,t ) • Has fixed isodynamic points X , at [0 , ∓√ / • Has fixed Brocard angle ω t = t/ • Has Brocard circle K t centered on X ,t = [0 , cos t −
22 sin t ] with radius ρ t = t −
12 sin t Proof.
Let T be the isosceles triangle defined by the vertices A = f , B = f and C = [0 , sin t t − ] ( t subscripts are omitted). Let vertex C be at the intersection ofthe line U f with the y axis. Applying Lemma 9 to this triangle it follows that: R ′ = 13 cos t − t −
84 sin t , ш ′ = 2 − cos t + 2sin t Invert the expressions for R ′ , ш ′ in Theorem 1 to obtain the anti Brocard triangle T ∗ of T , inscribed in Γ t . By construction, C = X of T ∗ and the pair ( E t , Γ t ) is aBrocard porism. (cid:3) Remark . Any Brocard porism is a similarity image of a porism B t defined inTheorem 4.Referring to Figure 7 (right): Corollary 11 (Brocard Circle Containment) . Let s, u ∈ [0 , π/ and s > u . Thenthe corresponding Brocard circles K s ⊂ K u . N INFINITE, CONVERGING, SEQUENCE OF BROCARD PORISMS 13
Figure 6.
Left:
A continuous family of inellipses (gray) is shown with foci (green dots) slidingalong two ◦ circular arcs (red) delimited by P , X and U , X . A particular one ( t = 36 ◦ )is highlighted (thick blue), with foci Ω , Ω . A Brocard porism is formed when E t is paired withcircle Γ t (blue). An isosceles 3-periodic is shown (blue) interscribed in the pair. Also shown is thefixed Brocard circle (dashed blue). Right:
The family of circles Γ t (gray) is nested and convergesto X . Video Proof.
Combining the the expressions for center and radius of K t and in Theorem 4the claim is true if and only if: sin( s − u ) + sin u ≥ sin s. where s, u ∈ [0 , π ] with s > u . As the above inequation is equivalent to sin( s − u )(1 − cos u ) + sin u (1 − cos( s − u )) ≥ follows the result. (cid:3) Theorem 5 (Embedding) . The family of second Brocard triangles of B t are the3-periodics of a distinct porism B t ′ , t ′ > t , such that: cot t ′ = ш ′ = 4 − t + cos 2 t t − sin 2 t = ш + 32 шProof. By Theorem 4 the inellipse which yields ш t is given by t = h ( ш t ) =tan − ( ш t ш t − ) . From Theorem 1 obtain that ш ′ = g ( ш t ) = ( ш t + 3) / (2 ш t ) . Takingthe composition ( h ◦ g ◦ h − )( t ) leads to the result. (cid:3) Proposition 7.
The circles K t are perpendicular to C and C Proof.
Let C , : (cid:18) x ± (cid:19) + y − K t : x + (cid:18) y − cos t −
22 sin t (cid:19) − (cid:18) t −
12 sin t (cid:19) = 0 The intersection of the circles C , and K t are the points p , ∓ = (cid:20) ∓ t − − t ) , − t − t (cid:21) , p , ∓ = (cid:20) ∓ ( 12 − cos t ) , sin t (cid:21) Direct calculations show that h∇ C ( p i, ∓ ) , ∇ K ( p i, ∓ ) i = 0 . (cid:3) Remark . For any B t , the midpoint X of P U is the inverse with respect to Γ t of the symmedian point X ,t .Let Z t denote the lower vertices of E t . Setting the derivative of b in Theorem 4to zero obtain: Corollary 12.
The minor semi-axis b (resp. Z t ) of E t reverses direction of mo-tion at t b = cos − (3 / ≃ . ◦ (resp. t = tan − (4 / ≃ . ◦ ). Furthermore ≤ b ≤ / . Note that the major semi-axis a and the y-coordinate of the upper vertex of E t are monotonically decreasing. Proposition 8.
The envelope ξ of the family E t is given by: ξ ,t , ξ ,t = (cid:20) ± √ t − √ cos t + 1 , − sint cos t + 1 (cid:21) and this is contained in the ellipse given by x + y − Furthermore, the isodynamic points X , X are at its foci.Proof. The ellipse E t is parametrized by Γ( u, t ) = [ a ( t ) cos u, b ( t ) sin u ] + [0 , − sin ( t )] The envelope is the solution to Γ( u, t ) = Γ t ( u, t ) = 0 which leads to the claim. (cid:3) Remark . At t > t = cos − ( ) = tan − ( ) the family E t is nested and so theenvelope is empty for t > t . See Fig. 7.
Proposition 9.
When t < tan − ( ) the Brocard Circle K t intersects the Brocardinellipse E t at its envelope ξ ,t and ξ ,t .Proof. The Brocard circle passes through the point [0 , − sin t − cos t ) ] and it is tangentto the Brocard inellipse at the point [0 , − when tan t = 4 / . (cid:3) Proposition 10.
The major semi-axis a is concave and monotonically-decreasingwith a (0) = 1 , a ( π ) = 0 . The minor semi-axis b is concave with a global maximum attained at t = cos − ( ) and b (0) = b ( π ) = 0 . The eccentricity ε ( t ) = c ( t ) a ( t ) is aconcave function with ε (0) = 1 , ε ( π ) = 0 .Proof. Direct analysis from Theorem 4. (cid:3)
N INFINITE, CONVERGING, SEQUENCE OF BROCARD PORISMS 15
Figure 7.
The envelope (thick black) of inellipses E t (gray) is an ellipse (only bottom half shown)whose foci are the isodynamic points X , X common to all families. At t = cos − (3 / , theporism is such that the Brocard circle (dashed blue) is tangent to E t (solid blue), X is at theenvelope’s lower vertex, and the lower vertex of E t reverses direction of motion. A non-isosceles3-periodic (dashed blue) is also shown. The locus (orange) of points at which the 3-web formed bythe family of inellipses E t is perpendicular to the family of Brocard circles K s (taken as independentfamilies) is a quartic containing the isodynamic and Beltrami points; see Remark 9. Let V l ( t )Γ( − π , t ) = [0 , − b ( t ) − sin t ] denote the lower vertex of E t . Corollary 13. V l ( t ) moves non-monotonically along the Brocard axis and convergesto X = [0 , − √ ] . At t = tan − ( ) it attains a global minimum [0 , − .Proof. Direct analysis from Theorem 4. (cid:3)
Remark . At t = tan − ( ) , ш = √ ≈ . . At t = cos − ( ) , ш = 2 , Remark . The family of ellipses E t is defined by the implicit differential equation x y dx − xy (cid:0) x − (cid:1) dx dy + (cid:0) x + 8 x + 4 y − (cid:1) dy = 0 The family of circles K t is given by the differential equation xy dx − (4 x − y + 3) dy = 0 Consequently, in the interior of the envelope, the family of circles K t and ellipses E t define a 3-web [1] which is singular at the coordinate axes and the boundary of the envelope. The orthogonal family to K t is the Apollonian pencil of circlespassing through the foci of the envelope. See [1] for the classification of other typesof 3-webs defined by circles and ellipses.Referring to Figure 7: Remark . The families of ellipses E t and circles K s (where t, s are independentparameters) are orthogonal, if and only if, x + 8 x + 4 y − . They are tangent, if and only if, xy = 0 .6. Conclusion
Above we describe various properties of a discrete sequence of Brocard porismsinduced by the second Brocard triangle, and how said sequence is embedded in acontinuous one. Videos mentioned in the text appear on Table 1.Exp Title Link (youtu.be/...)01 1st, 2nd, 5th, and 7th Brocard trianglesover the Brocard porism _bK-BCQv24A
02 Brocard porism and 2nd Brocard triangle
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03 Porism induced by family of 2nd Brocard triangles
MprJtB4UW9s
04 Infinite sequence of Brocard porismsinduced by the second Brocard triangle
Z3YlEbCFbnA
05 Continuous family of Brocard porisms jY_8zxBljuk
Table 1.
Illustrative animations, click on the link to view it on YouTube and/or enter youtu.be/ as a URL in your browser, where
is the provided string.
We would like to thank Peter Moses for his prompt and invaluable help with dozensof questions related to Brocard geometry, and Daniel Jaud for his generous andprecise editorial help. The second author is fellow of CNPq and coordinator ofProject PRONEX/ CNPq/ FAPEG 2017 10 26 7000 508.
Appendix A. Isosceles 3-Periodic
Consider an isosceles 3-periodic T = ABC in the Brocard porism, where AB istangent to E at one of its minor vertices. Let | AB | = 2 d and the height be h . Let ζ = d + h . Let the origin (0 , be at its circumcenter X . Its vertices will begiven by: A = (cid:20) − d, d − h h (cid:21) , B = (cid:20) d, d − h h (cid:21) , (cid:20) , ζ h (cid:21) Proposition 11.
The Brocard porism containing T as a 3-periodic is defined bythe following circumcircle K and Brocard inellipse E : K : x + y − R = 0 , R = ζ h E : − d h x − h (9 d + h ) ζy + 4 h (3 d + h )(3 d − h ) ζy − ( d − h )(9 d − h ) ζ = 0 N INFINITE, CONVERGING, SEQUENCE OF BROCARD PORISMS 17
Proof.
The proof follows from T , and isosceles 3-periodic. Recall that the Brocardinellipse is centered at X . Its perspector is X , i.e., it will be tangent to T wherecevians through X intersect it; see Figure 2. (cid:3) Proposition 12.
The semi-axes ( a, b ) of E are given by: ( a, b ) = (cid:18) d √ ζ d + h , d d + h (cid:19) Furthermore, the Brocard points, located at the foci of E are given by: Ω , Ω = (cid:20) ± d (3 d − h )9 d + h , d − h h (9 d + h ) (cid:21) Proof.
Follows directly from Proposition 11 computing the semi-axes and foci ofthe ellipse defined by E . (cid:3) Lemma 9. R = ζ h , ш = 3 d + h dh , sin ω = 2 dh p (9 d + h ) ζ Or inversely: d = − (cid:0) − ш + √ ш − (cid:1) R ш + 1 , h = ( ш + ш √ ш − R ш + 1 Proof.
Follows from Propositions 11 and 12. (cid:3)
Appendix B. Brocard Porism Vertices
Let d, h be the half base and height of T , respectively. Let ζ = d + h . Proposition 13.
Parametrics for the 3-periodic vertices in terms of t are givenby: (2) A = (cid:20) ζ h cos t, ζ h sin t (cid:21) B =[ b x , b y ] b x = − ζd (cid:0) dh cos t + (3 d + h ) sin t − d + h (cid:1) dh (3 d − h ) cos t − (9 d − h ) sin t + 9 d + 2 d h + h b y = ζ (cid:0) dh (3 d + h ) cos t − (9 d − d h + h ) sin t + 9 d − h (cid:1) dh (3 d − h ) cos t − (9 d − h ) sin t + 9 d + 2 d h + h C =[ c x , c y ] c x = − ζd (cid:0) − dh cos t + (3 d + h ) sin t − d + h (cid:1) (9 d − h ) sin t + 2 dh (3 d − h ) cos t − d − d h − h c y = ζ (2 dh (3 d + h ) cos t + (9 d − d h + h ) sin t − d + h )2 h (2 dh (3 d − h ) cos t + (9 d − h ) sin t − d − d h − h ) Proof.
Given a point A = R (cos t, sin t ) ∈ K consider the tangent lines to theBrocard inellipse E obtained in Proposition 11. The two other intersection pointsof these lines with K define the vertices B and C . A long and straightforwardcalculation with help of a CAS leads to the result stated. (cid:3) Appendix C. Further Relations
Expressions for X k , k = 6 , , , , were given above. They were obtainedby starting with the explicit calculation of the triangular center X i ( t ) over thefamily of 3-periodic orbits in the Brocard porism given in Proposition 12.To this end we make use of the trilinear coordinates f ( a, b, c ) :: of the triangularcenter X i [10] and its conversion to Euclidean coordinates expressed by X i = s f ( s , s , s ) A + s f ( s , s , s ) B + s f ( s , s , s ) Cs f ( s , s , s ) + s f ( s , s , s ) + s f ( s , s , s ) . Here s = | B − C | , s = | C − A | and s = | A − B | . Obtain the result by simplifi-cation via a CA. It is worth mentioning that all triangular centers considered arerational functions of ( d, h ) . Therefore, by Lemma 9 they will be rational functionsof ( ш , √ ш − . Lemma 10.
The distance between circumcenter X (respec. X ′ ) and symmedian X (respec. X ′ ) of T (respec. T ′ ) is given by | X − X | = R √ ш − ш , | X ′ − X ′ | = R ( ш − ш ( ш + 3) Proof.
Consider any triangle with circumradius R and Brocard angle ω . By theconstruction of the Brocard porism the family has fixed triangular centers X , X , X = ( X + X ) and X . As X ′ = X and X ′ = X the result followsfrom Lemmas 4 and 5. (cid:3) Convergence.
Recall that given a map f : U → U , U = ∅ , the positive orbitof point p is the set O + ( p ) = { p , f ( p ) , f ( f ( p )) , . . . , f n ( p ) , . . . } . Analogously,when f is invertible, the negative orbit is defined by O − ( p ) = { p , f − ( p ) , f − ( f − ( p )) , . . . , f − n ( p ) , . . . } . The future (resp. past) of an orbit is the closure of the positive (resp. negative)orbit and is denoted by the ω -limit set ω ( p ) (resp. α -limit set α ( p ) ). For anintroduction to more properties of these concepts see [5, Chapter 1]. Proposition 14 (Convergence) . Let f ( R, ш ) = ( p ( R, ш ) , q ( ш )) = R √ ш − ш , ш + 32 ш ! For any p = [ R , ш ] with R > and ш > √ , ω ( p ) is equal to [0 , √ and α ( p ) = [ ∞ , ∞ ] . Proof.
Define U = { ( R, ш ) : ш ≥ √ } . We have that U is invariant by f , i.e., f ( U ) ⊂ U. We observe that for ш > , q has a unique fixed point (0 , √ .For any ш ≥ √ , ω ( ш ) = √ for q since q ′ ( √
3) = 0 < (attractor). In fact aglobal attractor. As √ ш − ш < for ш > √ and p ( x ) < Rx it follows by a graphicanalysis that ω ( p ) = [0 , √ . The inverse of f is given by f − ( R, ш ) = ш + p ш − , √ R p y + √ ш − √ ш − ! N INFINITE, CONVERGING, SEQUENCE OF BROCARD PORISMS 19
A similar analysis shows that [0 , √ is a repeller of f − and that the positive orbitof p goes to infinity. (cid:3) Appendix D. Table of Symbols symbol meaning note
T, T ′ s , s , s sidelengths of T Γ , R circumcircle and circumradius of T E , a, b Brocard inellipse and semi-axes centered on X B Brocard porism with E and ΓΩ , Ω E ω, ш Brocard angle and its cotangent ш = cot ω K Brocard circle centered on X K second Brocard circle centered on X T isosceles triangle inscribed in Γ whose base is tangent to E at (0 , b ) d, h half-base and height of T ζ constant d + h P , U Γ -inverses of Ω , Ω C , C Beltrami Circles centered on P , U ,contain X and X X circumcenter (at origin) X = [0 , X symmedian point X , X isogonic points X , X isodynamic points X Brocard midpoint X center of K midpoint of X X X Midpoint of Beltrami points X X of T ′ K -inverse of X X X of T ′ Table 2.
Symbols used. Primed symbols in the text refer to the second Brocard triangle.
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