Loci of the Brocard Points over Selected Triangle Families
aa r X i v : . [ m a t h . M G ] S e p LOCI OF THE BROCARD POINTS OVERSELECTED TRIANGLE FAMILIES
RONALDO GARCIA AND DAN REZNIK
Abstract.
We study the loci of the Brocard points over two selected familiesof triangles: (i) 2 vertices fixed on a circumference and a third one whichsweeps it, (ii) Poncelet 3-periodics in the homothetic ellipse pair. Loci obtainedinclude circles, ellipses, and teardrop-like curves. We derive expressions forboth curves and their areas. We also study the locus of the vertices of Brocardtriangles over the homothetic and Brocard-poristic Poncelet families.
Keywords
Poncelet, Brocard, Homothetic, Porism, Locus
MSC Introduction
The Brocard points Ω and Ω [8] are two unique interior points through whichCevians at a special angle ω to the sides concur; see Figure 1. In triangle parlance,they are known as a bicentric pair [9]. Here we study the loci of Ω , Ω over a familyof triangles over two selected families of triangles, to be sure: • Circle-Mounted: two vertices fixed at points on a circumference (or ellipseboundary) and a third one which slides over it. • Homothetic Pair: Poncelet 3-Periodics interscribed in a homothetic, con-centric pair of ellipses. This family conserves the Brocard angle [13].
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 RONALDO GARCIA AND DAN REZNIK Under the above Brocard loci include a circle, teardrop-curves, and variations.We derive explicit expressions for the loci and study their area ratios with respectto the generating curves.We also study the locus of vertices of the Brocard Triangles (there are 7 of them,see [7]) under two families: homothetic and Brocard-poristic (whose Brocard pointsare stationary [1]). In the former case the only the vertices of the First Brocardis an ellipse, and in the latter 4 out of 7 describe circles, both results stemmingdirectly from known homotheties of said triangles.1.1. Related Work.
Ferréol describes a construction for a right strophoid as thelocus of the orthocenter of a family of triangles with two fixed vertices and a thirdone revolving on a circumference [3]. Odehnal has studied loci of triangle centersfor the poristic triangle family [11]; similarly, Pamfilos proves properties of thefamily of triangles with fixed 9-point and circumcircle [12]. We have studied theloci of triangle centers for 3-periodics in the elliptic billiard (confocal Poncelet pair),identifying a few centers whose loci are elliptic [5], and built and interactive app tovisualize loci of centers of several triangle families [2].Below when referring to triangle centers we adopt Kimberling’s X k notation[10]. 2. Circle-Mounted Triangles
Let a family of triangles be defined with two vertices V , V stationary withrespect to a circle of radius a (say centered at the origin) and a third one P ( t ) which executes on revolution over the the circumference, P ( t ) = a [cos t, sin t ] .Referring to Figure 2 (left): Proposition 1.
The locus of Ω (resp. Ω ) with V = (0 , and V = (0 , a ) is acircle of radius a (resp. a teardrop curve) of area πa (resp. πa ).Proof. In this case we have that: Ω ( t ) = a (cid:20) cos t − t , − a sin t − t (cid:21) Ω ( t ) = a (cid:20) t − sin 2 t − t , t + cos 2 t − t (cid:21) (cid:3) Remark . The above loci intersect at a [ ±√ / , / ; along with V = (0 , a ) theydefine an equilateral. This stems from the fact that when P ( t ) = a [ ±√ / , / , V V P ( t ) is equilateral and the two Brocard points coincide at the Barycenter X .Referring to Figure 2 (right): Proposition 2.
The locus of Ω and Ω with V = (0 , a ) and V = ( a, are apair of inversely-identical , skewed teardrop shapes with the following equations andareas: The 7th Brocard was invented by one author and Peter Moses during this research: its verticesare the intersections of cevians through X with the Brocard Circle. The inspiration was the 2ndBrocard whose vertices lie at intersections of cevians through X with said circle [14, SecondBrocard Triangle]. Identical modulo inverse similarity [14, Inversely Similar].
OCI OF THE BROCARD POINTS OVER SELECTED TRIANGLE FAMILIES 3
Proof.
We have that: Ω ( t ) = a (cid:20) sin t + cos t − sin t (sin t −
2) cos t − t + 3 , − cos t (sin t −
2) cos t − t + 3 (cid:21) Ω ( t ) = a " − sin t (cos t −
2) sin t − t + 3 , (cid:0) cos t − cos t + sin t (cid:1) (sin t −
2) cos t − t + 3 Defining D ( x, y ) = ( y, x ) , the reflection abott the diagonal, it follows that Ω ( t ) = ( D ◦ Ω )( t − π ) . (cid:3) In Figure 3 we show the shape of the locus varies in a complicated way when V = (0 , a ) and V = ( x, , with ≤ x ≤ a .Referring to Figure 4 (left), Robert Ferréol has kindly contributed [4]: Proposition 3.
With V = ( − a, and V = ( a, , the loci of the Brocards are apair of inversely-identical teardrop shapes whose areas are πa / √ . The one witha cusp on V is given by the following quartic: (1) x − x + 2 y x + 2 x − y x − y + 4 y = 0 Proof. Ω ( t ) = (cid:20) − a cos 2 t − a cos t + a cos 2 t − , − a sin 2 t − a sin t cos 2 t − (cid:21) Ω ( t ) = (cid:20) − a cos t + a cos 2 t − a cos (2 u ) − , a sin 2 t − a sin t cos 2 t − . (cid:21) Let R ( x, y ) = ( − x, y . Then Ω ( t ) = ( R ◦ Ω )( t ) . The implicit form of Ω is givenby B ( x, y ) = a ( a − ax − y ) + 2 ax ( x + y ) − ( x + y ) = 0 . Analogously, B ( x, y ) = B ( − x, y ) = 0 is the implicit form of Ω .The area of the region bounded by Ω i is given by R Ω i xdy − ydx . It follows that A (Ω i ) = √ πa . (cid:3) Figure 4 (right) depicts the loci of Ω and Ω with P ( t ) on an ellipse with semi-axes ( a, b ) and with V = ( − a, and V = ( a, . These are a pair of symmetricteardrop curves whose complicated parametric equations we omit. Proposition 4.
The locus of Ω and Ω with V = ( x , , | x | ≤ a , V = ( − a, and V = ( a cos t, a sin t ) are a pair of singular teardrop curves with the followingareas: A = 4 ( x + a ) a π (3 a + x ) p a + x A = (cid:0) a − ax + x (cid:1) ( x + a ) a π (3 a + x ) p a + x When x = a , the ratio of A and A by the area of the circle a π both reduceto / √ ≃ . . Proof.
The above is obtained with direct integration and simplification with a com-puter algebra system (CAS). (cid:3)
RONALDO GARCIA AND DAN REZNIK
Figure 2.
Left: V and V are affixed to the center and top vertex of the unit circle and athird one P ( t ) revolves around the circumference. The locus of the Brocard points Ω , Ω are acircle (red) and a teardrop (green) whose areas are 1/9 and 2/9 that of the generating circle. Thesample triangle (blue) shown is equilateral, so the two Brocard points coincide. Notice the curves’two intersections along with the top vertex form an equilateral (orange). Right: V , V are nowplaced at the left and top vertices of the unit circle. The Brocard points of the family describe toinversely-similar teardrop shapes. Video, Live Homothetic Pair Loci
Consider an origin-centered ellipse with semi-axes ( a, b ) and an internal concen-tric, axis-aligned one with semi-axes ( a ′ , b ′ ) = ( a/ , b/ . This pair is associatedwith a 3-periodic Poncelet porism since the condition a ′ /a + b ′ /b = 1 is satisfied [6].Referring to Figure 5: Proposition 5.
The loci of the Brocard points over 3-periodics in the homotheticpair are ellipses E and E which are reflected images of each other about eitherthe x or y axis. Furthermore these are concentric and homothetic to the ellipses inthe pair.Proof. The loci are given by E ( x, y ) = (cid:0) a + 6 a b + 3 b (cid:1) x a ( a − b ) + (cid:0) a + 6 a b + 7 b (cid:1) y b ( a − b ) − √ (cid:0) a + b (cid:1) xyab ( a − b ) − E ( x, y ) = (cid:0) a + 6 a b + 3 b (cid:1) x a ( a − b ) + (cid:0) a + 6 a b + 7 b (cid:1) y b ( a − b ) + 4 √ (cid:0) a + b (cid:1) xyab ( a − b ) − The angle θ between the axes of ellipses E and E is given by tan θ = 4 √ a + b ) ab a + 2 a b + 3 b . (cid:3) In no other concentric Poncelet pairs studied so far (poristic, incircle, inellipse,dual, confocal) is the locus of either Brocard point an ellipse.
Remark . At a/b = √ the elliptic loci of the Brocard points over the homotheticfamily are internally tangent to the inner ellipse. Remark . At a/b ≃ . the elliptic loci of the Brocard points over the homotheticfamily intersect the y axis at b/ , i.e., at the top vertex of the caustic. OCI OF THE BROCARD POINTS OVER SELECTED TRIANGLE FAMILIES 5
Figure 3.
Shape of loci of Brocard Point Ω (red) and Ω (green) for V fixed at (0 , , as V movesfrom the origin along the x axis toward (1 , . The loci are obtained over a complete revolutionof P ( t ) on a unit circle (black). Top left: V = (0 , , the locus of Ω (resp. Ω ) is a perfectcircle (resp. a teardrop curve) of / (resp. / ) the area of the external. Bottom right: when V = ( a, the two loci are inversely-similar copies of each other, whose areas are / √ ≃ . that of the circle. Video, Live RONALDO GARCIA AND DAN REZNIK Figure 4.
Left:
With antipodal V and V and P ( t ) revolving on the circumference, the lociof the Brocards are symmetric teardrops whose area are / √ that of the circle. Right: . With V , V at the major vertices of an ellipse of axes ( a, b ) , and P ( t ) revolving on its boundary, the theBrocard loci (red and green) are still symmetric (though stretched) teardrop shapes. In this case a/b = 1 . . Live Figure 5.
The 1d Poncelet 3-periodic family interscribed in the homothetic pair conserves sum ofsquared sidelengths, area, and Brocard angle ω [13]. The loci of the two Brocard points Ω and Ω are tilted ellipses (red and green) of aspect ratio equal to those in the pair Video. The locus(dashed orange) of the vertices of the first Brocard triangle (orange) is an axis-aligned ellipse alsohomothetic to the pair.Video, AppOCI OF THE BROCARD POINTS OVER SELECTED TRIANGLE FAMILIES 7 Figure 6.
Construction for the First Brocard Triangle (orange) taken from [14, First BrocardTriangle]. It is inversely similar to the reference one (blue), and their barycenters X are common.Its vertices B , B , B are concyclic with the Brocard points Ω and Ω on the Brocard circle(orange). In Appendix A we derive a few properties of the Brocard circle with respect tothe homothetic family. 4.
Brocard Triangles
Consider a triangle T = P P P with Brocard points Ω and Ω . Referring toFigure 6: Definition 1 (First Brocard Triangle) . The vertices P ′ , P ′ , P ′ of the First BrocardTriangle T are defined as follows: P ′ (resp. P ′ , P ′ ) is the intersection of P Ω (resp. P Ω , P Ω ) with P Ω (resp. P Ω , P Ω ).Know properties of the T include that (i) it is inversely similar to T , (ii) itsbarycenter X coincides with that of the reference triangle, and (iii) its verticesare concyclic with Ω , Ω , X , and X on the Brocard circle [14, Brocard Circle],whose center is X .4.1. Homothetic Pair.
Referring to Figure 5:
Proposition 6.
Over 3-periodics in the homothetic pair, the locus of the verticesof T is an axis-aligned, concentric ellipse, homothetic to the ones in the pair andinterior to the caustic. Its axes are given by: a ′ = a ( a − b )2( a + b ) , b ′ = b ( a − b )2( a + b ) Proof.
The locus must be an ellipse since T is inversely similar to the 3-periodicswhose vertices are inscribed in an ellipse and their barycenters coincide. A vertexof the Brocard triangle is parametrized by x a ′ + y b ′ = 1 It can be shown a ′ < a/ and b ′ < b/ therefore the locus is interior to thecaustic, i.e., the stationary Steiner inellipse. (cid:3) RONALDO GARCIA AND DAN REZNIK
Since homothetic 3-periodics conserve area [13], so must T (inversely similar).Its area can be computed explicitly: Remark . Over 3-periodics in the homothetic pair, the area of T is invariant andgiven by √ ab (cid:0) a − b (cid:1)
16 ( a + b ) Corollary 1.
The similarity ratio of homothetic 3-periodics to the T is invariantand given by a + b ) a − b · Vertices of the T over the homothetic family. Let the P P P be thevertices of a 3-periodic in the homothetic pair, and P ′ P ′ P ′ those of T . These aregiven by: P ′ = k R x P P ′ = k R x P P ′ = k R x P where k = a − b a + b ) and R x ( x, y ) = ( − x, y ) is a reflection.4.3. Brocard Porism.
This is a 3-periodic family inscribed in a fixed circumcircleand a fixed inellipse, known as the Brocard Inellipse [14, Brocard Inellipse]. Aremarkable property of this family is that the Brocard points are stationary at thefoci of said inellipse [1].Given the axes ( a, b ) of the latter, we have shown elsewhere that the circumcenter X , circumradius R , and (conserved) Brocard angle ω of the family are given by[13]:(2) X = [0 , − cδ b ] , R = 2 a b , cot ω = δ b where δ = √ a − b .Let T k denote the nth-Brocard triangle, k = 1 , · · · , , as defined in [7]. Referringto Figure 7 Remark . For k = 1 , , , , the locus of vertices of T k trace out the Brocard circle.This stems from the fact that by construction, T , T , T are inscribed in theBrocard circle (their circumcenter is X and T is homothetic to the reference oneand its circumcenter is X [10]. For no other Poncelet families and/or Brocardtriangle combinations have we been able to identify conic loci for the BrocardTriangle vertices.As before, Ω , Ω denote the Brocard points of the 3-periodics. Let Ω j (resp. Ω j ) denote the first (resp. second) Brocard point of T j . Also let X ji denote the X i center of T j . The following observations are about the 3-periodic family in theBrocard porism. OCI OF THE BROCARD POINTS OVER SELECTED TRIANGLE FAMILIES 9
Figure 7.
A 3-periodic (blue) in the Brocard Porism is shown inscribed in a circle and circum-scribed about the Brocard inellipse (both black). Its Brocard points Ω and Ω are stationary atthe foci of said inellipse. The First, Second, and Seventh Brocard Triangle are shown inscribed inthe Brocard circle (green). The Fifth Brocard Triangle (orange)is homothetic about the 3-periodicand therefore its locus will also be a circle (not shown). Video, Live Observation 1. Ω and Ω are stationary. Furthermore the triples (Ω , X , Ω ) and (Ω , X , Ω ) are each collinear. Remarkably, the above implies a “russian doll” nesting of Brocard porisms com-posed of the original family and then successive Second Brocards, each inhabitingits own private porism, see this video.
Observation 2.
Both Ω and Ω move along the same circle C . Observation 3.
Both Ω and Ω move along the same circle C . Observation 4.
The locus of Ω and Ω are two distinct circles C and C ′ . Conclusion
A few questions are posed to the reader: • Are there other Poncelet ellipse pairs, concentric or not, whose Brocardpoints of their 3-periodics trace out ellipses? (so far he have found none) • In the homothetic pair are there other derived triangles (besides the FirstBrocard), non-homothetic to the 3-periodics, whose vertices trace out con-ics?A list of animations of some of the results above appears on Table 1. T j , j = 1 , , ,
05 * n/a Russian-Doll Nesting of Second Brocard Triangles
Table 1.
Experimental animations. Click on the * to see it as a YouTube video and/or a browser-based simulation.
We would like to thank Robert Ferréol for his Mathcurve portal (it inspired thiswork) as well as early derivations of loci. Mark Helman has kindly helped us withexperiments and research around the Brocard loci. Bernard Gibert and Peter Mosesfor helping us usher the 7th Brocard Triangle into existence. The first author isfellow of CNPq and coordinator of Project PRONEX/ CNPq/ FAPEG 2017 10 267000 508.
Appendix A. Brocard Circle of the Homothetic Pair
Recall the Brocard circle is the circumcircle of the T [14, Brocard Circle]. Corollary 2.
Over 3-periodics in the homothetic pair, the ratio of areas of thecircumcircle of 3-periodics to that of the Brocard circle is invariant and given by a + b ) ( a − b ) · Proof.
The result follows from Corollary 1. Straightforward calculations yield thatthe circumradius of 3-periodics is given by: R = − ( a − b ) cos(6 t )32 a b + ( a + b )( a + 14 a b + b )32 a b . (cid:3) Proposition 7.
Over the homothetic family, the Brocard circle is given by: E ( x, y ) = 4 (cid:0) a + b (cid:1) x a ( a − b ) + 4 (cid:0) a + b (cid:1) y b ( a − b ) − References [1] Bradley, C., Smith, G. (2007). On a construction of Hagge.
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