I’m going to assume that you love beautiful things and are curious to learn about them. The only things you will need on this journey are common sense and simple human curiosity. –Paul Lockhardt, “A mathematician’s lament” (2009)
Below we provide links to some of the artifacts (webpages, images, videos, applets, code) from ongoing experiments with triangles, billiards, ellipses, and other marvellous geometric objects. The fun and amazement with their beauty truly never ends.
Our Main Pages
Invariants of 3 and NPeriodics in an Elliptic Billiard, with R. Garcia and J. Koiller.
Loci of NPeriodics and their Derived and Internal Triangles
Envelopes and Evolutes: beautiful curves enveloped by pairs \(X_i,X_j\) of Triangle Centers.
new Loci of Centers of EllipseMounted Triangles: astounding loci generated by triangle centers when a vertex slides along and ellipse and the other two vertices are held fixed. With M. Helman.
new Artsy Loci of EllipseMounted Triangles: long, short
new ExtremalArea Pedal and Antipedal Triangles: a discussion on maximal and minimal pedals and antipedals to a triangle. With M. Helman.
new Loci of 3periodics in the Elliptic Billiard: Why so many ellipses?: semiaxes and fit errors for 29 triangles centers (out of the first 100 in [1]) whose loci under 3periodics in the elliptic billiard are ellipses. With R. Garcia and J. Koiller.
Videos N=3
All videos are available as a single playlist.
Original 2011
Title

sound

Kimberling Centers

Year

urls

3Periodic trajectories


–

2011

v1, v2, v3

Locus of incenter is elliptic for family of 3periodics


1

2011

v1

Locus of the incircle touchpoints is a higherorder curve


1

2011

v1

Early Results
Title

sound

Kimberling Centers

Year

urls

Elliptic Loci of X(1) to X(5) and Euler Line


1,2,3,4,5

2019

v1

Loci of Vertices of Medial, Intouch and Feuerbach Triangles is not elliptic

T

1,5

2019

v1

Mittenpunkt is stationary at center of billiard


9

2019

v1, v2

Feuerbach Point Sweeps Billiard and its AntiComplement and Extouch Points sweep caustic

T

1,2,5,11,100

2019

v1

Conservation of Sum and Product of Cosines

T

–

2019

v1

N=3 Loci
Title

sound

Kimberling Centers

Year

urls

3Periodics and Derived Triangles


–

2019

v1

Elliptic Loci of X(1) to X(5) and Euler Line


1,2,3,4,5

2019

v1

Locus of several triangular centers is elliptic


1,2,3,4,5

2019

v1

Locus of vertices of Feuerbach Triangle is nonelliptic


–

2019

v1

NonElliptic loci of vertices of Medial, Intouch and Feuerbach triangles

T

1,5,11

2019

v1

Locus of Bevan Point X(40) is similar to billiard


40

2019

v1

Locus of Bevan Point X(40) identical to billiard when a/b=golden ratio


40

2020

v1

Locus of X(59) has 4 selfintersections


59

2020

v1

Anticomplementary triangle intouchpoints


1,2,7,9,100,144

2019

v1, v2

Anticomplementary, Medial Triangles and the Intouch Triangle


1,2,7,9,11,100,142,144

2019

v1, v2

The locus of X(140) is a circle over 3periodics in the Elliptic Billiard

T

3,5,140

2020

v1

Circular loci for X(140) and X(547) over 3periodics in the Elliptic Billiard

T

3,2,5,40,547

2020

v1

Feuerbach
Title

sound

Kimberling Centers

Year

urls

Locus of Feuerbach point, its anticomplement and three extouchpoints

T

1,2,5,11,100

2019

v1, v2, v3

Locus of Excentral and Anticomplementary Triangles and Objects


1,2,5,11,100

2019

v1, v2

Circumconics
Title

sound

Kimberling Centers

Year

urls

The X(1) and X(2)centered circumellipses


1,2,9

2019

v1, v2

Locus of Intersection of X(1) and X(2)centered circumellipses


1,2,75,77,100,190

2019

v1

The X(100)centered Excentral Jerabek Hyperbola


1,9,100

2019

v1

The Feuerbach and Excentral Hyperbolas


1,4,7,8,9,11,40, 100,144,1156

2019

v1, v2

The Jerabek Hyperbola and Circumbilliard of the Excentral Triangle


1,9,40,100, 168,1156

2019

v1

Peter Moses’ Points on the X(9)centered circumellipse


88, 100, 162, 190, 651, 653, 655, 658, 660, 662, 673, 771, 799, 823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599, 4604, 4606, 4607, 8052, 20332, 23707, 24624, 27834, 32680

2019

v1, v2

Invariants of the X(1)centered circumellipse


1,3,100

2019

v1

Invariants of the Steiner Circum and Inconics


2,190

2019

v1

The Yff Parabola, Contact Triangle & Loci of Vertex and Axis Foot


4,9,101,190,3234

2020

v1, v2

Every triangle has a unique Circumbilliard


9

2019

v1

Circumbilliard of anticomplementary triangle


1,2,5,7,11,100

2019

v1

Orthic Circumbilliard & Locus of Its Mittenpunkt


4,6,9

2020

v1

Circumbilliards of Triangles Derived from 3Periodics


7,8,142,168

2020

v1

Feuerbach and Excentral Jerabek Circumhyperbolas: Invariant Focal Length Ratio

T

1,4,9,11,40,100,1156

2020

v1, v2

Inconics
Title

sound

Kimberling Centers

Year

urls

Excentral MacBeath Inconic: Invariant Aspect Ratio

T

1,3,40,1742

2020

v1

X(3)Centered Excentral Inconic: Invariant Aspect Ratio

T

3,40,69,2951

2020

v1

Convex Combinations
Title

sound

Kimberling Centers

Year

urls

Barycenter with Median, and Incenter with Intouchpoint


1,2

2019

v1

Orthocenter with one altitude foot, and Circumcenter with median


3,4

2019

v1

Excenter and its corresponding Extouch point


–

2019

v1

Orthic Phenomena
Title

sound

Kimberling Centers

Year

urls

Locus orthic triangle’s incenter is a 4arc ellipse


4

2019

v1

Locus of orthocenter, orthic orthocenter, incenter, and orthic orthic’s incenter


1,4

2019

v1

Excentral of Orthic for Acute and Obtuse Triangles


4

2019

v1

Stationary Circles
Title

sound

Kimberling Centers

Year

urls

Cosine Circle of Excentral Triangle is Stationary


9,40

2019

v1

Locus of Intersection of AntiTangents is Stationary Circle


9

2019

v1

Intersections of Excentral Triangle and its Reflection is a Circle


9

2019

v1

Unrolled 3Periodics
Title

sound

Kimberling Centers

Year

urls

Fixed central billiard


–

2020

v1

Pin P1 and n1


–

2020

v1

Pin P1 at origin


–

2020

v1

Pin P_1 at origin and P1’’ vertically above it


–

2020

v1

Poncelet Family
Title

sound

Kimberling Centers

Year

urls

Poncelet Triangle Inscribed in Ellipse and Circumscribed in Circle


1,2,3,4,5,11,100

2019

v1

Poncelet Family of Triangles over the Family of N=3 Caustics


1,2,3,4,5,11,9,100

2019

v1

Pencil of N=3 Poncelet Ellipse Pairs: Loci of Triangular Centers


1,2,3,4,5,6,7,8,9,11,100

2019

v1

Three Geometers Walk into a Bar: the 3periodic PonceletSteiner family has invariant Brocard angle.

T

2

2020

v1

Poristic
Title

sound

Kimberling Centers

Year

urls

Chapple’s Porism from (1746) and Weaver (1927) and Odehnal (2011) Invariants

T

1,3,40,1155

2020

v1

Circumbilliard of the Poristic Triangle Family: Invariant Aspect Ratio

T

1,3,9,40

2020

v1

Poristic Triangle Family and the Amazing Invariant Excentral X3Centered Inconic

T

1,3,9,40,100

2020

v1

Simson Lines from X100 and Excentral Medials are Parallel to L(X1,X3).

T

1,3,40,100

2020

v1

X1Centered Circumconic & X40Centered (Excentral) Inconic: Identical Invariant Axes

T

1,3,40,100

2020

v1, v2

Loci of center and foci of the Circumbilliard to the Poristic Family are circles.

T

1,3,9,40,100

2020

v1

Aspect Ratios of X10 and Excentral X5Centered Circumconics are Invariant & Equal

T

1,3,10,40,100

2020

v1

Invariant aspect ratios for the Circumbilliard and Excentral X6Ctr Circumconic

T

1,3,9,40,100

2020

v1

SidebySide View of Poristic and 3Periodic Families

T

1,3,9,40,100

2020

v1

Feuerbach and Excentral Jerabek Hyperbolas to Poristic Family have invariant focal length ratio


1,3,9,11,100

2020

v1

Reference & Excentral Simson Lines have fixed points and are Orthogonal!

T

1,3,40

2020

v1, v2

Misc
Title

sound

Kimberling Centers

Year

urls

Loci of Outer Napoleon Equilateral Construction


13

2019

v1, v2

Conservation of Sum and Product of Cosines


–

2019

v1

The Miquel Point of the Extouch and Excentral Triangles


40

2019

v1, v2

An invariant in the parabolic pair associated with the N=3 family


–

2020

v1

Nonmonotonic X(88) and the X(1)X(100) envelope


1,88,100

2020

v1

The Thomson Cubic of 3periodics


1,2,3,4,6,9

2020

v1

Locus and elliptic envelope of excircle tangents’ hexagon (side touchpoints)

T

–

2020

v1

SixPoint Conic passes through Sideline Tangents to Excircles

T

–

2020

v1

Elliptic Billiard 3Periodics: Invariants of the Focal Hyperbola

T

–

2020

v1

Isogonal and Isotomic
Title

sound

Kimberling Centers

Year

urls

Antiorthic Axis and 5 points on the Billiard


1,6,9,44,88,100

2019

v1

Isotomic and Isogonal Conjugates of Billiard with respect to the 3periodic family


1,9,144

2019

v1

Envelopes
Title

sound

Kimberling Centers

Year

urls

Envelope of Antiorthic and Gergonne Lines


9,44,1155,857,908

2020

v1

Evolute of Elliptic Billiard and Envelope of X(1)X(5)


1,5,4,9,

2020

v1

Envelope of 3Periodic Vertex with Triangle Center


1,2,3,4,5,6,7,8,10,11,12,20

2020

v1

Evolute Triangles of P1(t) with X(i)


1,3,5,20

2020

v1

Elliptic Envelope of P1(t) with P1(t+pi/2)


–

2020

v1

Envelope of 3Periodic P1 and reflected P2 is Elliptic

T

–

2020

v1

The BatEnvelope of X(48) and X(37143)

T

48,37143

2020

v1

Envelopes of Sides of Derived Triangles

T

–

2020

v1

Envelope of Simson Lines from X100 and X99 to two N=3 Poncelet Families

T

2,3,9,99,100

2020

v1

Swans
Title

sound

Kimberling Centers

Year

urls

Motion of X(88) with respect to collinear X(100) and X(1)


1,88,100

2020

v1

Dance of the Swans: X(88) and X(162)

T

9,88,162

2020

v1, v2

Pedal Invariants
Title

sound

Kimberling Centers

Year

urls

Area Invariants of Pedal and Antipedal Polygons

T

–

2020

v1

EllipseMounted
Title

sound

Kimberling Centers

Year

urls

EllipseMounted Triangles: Elliptic locus of the Orthocenter X(4) and suprising area invariance!

T

4

2020

v1

CircleMounted Triangles: Surprising Loci of the Brocard Points

T

–

2020

v1

Poncelet
Title

sound

Kimberling Centers

Year

urls

An N=3 Poncelet family (outer circle, inner ellipse) equivalent to Poristic Excentrals

T

3,4,5

2020

v1

Between an Ellipse and a Concentric Circle: Poncelet 3Periodics Identical to Poristic Triangles.

T

1,3

2020

v1

Between a Circle and a Concentric Ellipse: Poncelet 3Periodics Identical to Poristic Excentrals.

T

3,4,5

2020

v1

3Periodics in a Concentric Homothetic Poncelet Pair: Circular Loci of four Triangle Centers

T

2,13,14,15,16

2020

v1

3Periodics in a HomotheticRotated Poncelet Pair: stationary orthocenter and loci of X107 and X122

T

4,107,122

2020

v1

Poncelet 3Periodic Invariants (Outer Circle, Inner Concentric Ellipse) of the NinePoint Center II

T

3,5,4

2020

v1

Isodynamic Pedals and Isogonic Antipedals: Equilaterals with Constant Area in the Homothetic Pair

T

2,13,14,15,16,395,396,5463,5464

2020

v1

Brocard
Title

sound

Kimberling Centers

Year

urls

Poncelet 3Periodics of Homothetic Pair: Elliptic Loci of Brocard Pts + Vertices of 1st Brocard Tri

T

2

2020

v1

It takes 2 to tango: BrocardPoncelet Porism, stationary Brocard Points and invariant Brocard Angle

T

3,6,39,182

2020

v1

Joined at the hip: Brocard Porism, Steiner Ellipses, and the Homothetic Poncelet Pair

T

2,3,6,182

2020

v1

The Poncelet Homothetic Pair contains an AspectRatio Invariant Brocard Inellipse

T

2,39

2020

v1

Brocard Porism: Locus of 1st, 2nd, 5th, and 7th Brocard Triangles’ Vertices are Circles

T

3,6,39,182,9821

2020

v1

RussianDoll nesting of Brocard porisms courtesy of the second Brocard triangle

T

6,39,182

2020

v1

Rusiandoll nesting of Brocard porisms: concyclic sequence of Brocard points and the Beltrami points

T

6,39,182

2020

v1

Brocard Porism: equilateral Isodynamic Pedals have invariant area ratio + circular centroidal locus

T

3,6,15,16,39,182

2020

v1

Continuous Family of Brocard Porisms with Stationary Isodynamic Points X15 and X16

T

3,6,15,182

2020

v1

The Family of Second Brocard Triangles in the Brocard Porism

T

3,6,15,39,182

2020

v1

Brocard Porism: Family of Second Brocard Triangles is a second Brocard Porism

T

3,6,15,39,182,39498

2020

v1

Inversive Poncelet
Title

sound

Kimberling Centers

Year

urls

FocusInversive N=3 Family in the Elliptic Billiard: Pascal LimaçonInscribed Billiard Triangles!

T

9,7

2020

v1, v2

FocusInversive Polygons’ EquiArea Pedal Polygons (wrt foci)

T

–

2020

v1

EllipseInscribed Triangles
Title

sound

Kimberling Centers

Year

urls

Loci of EllipseInscribed Triangles I: Basic Phenomena

T

2,3,4,5,381

2020

v1

Loci of EllipseInscribed Triangles II: X() slides merrily along the Euler line

T

2,4

2020

v1

Loci of EllipseInscribed Triangles III: family of V1V2 parallels causes rigid locus translation

T

–

2020

v1

Loci of EllipseInscribed Triangles IV: Multiple Loci Over Parallel V1V2

T

–

2020

v1

Loci of EllipseInscribed Triangles V: Circular Loci if V1V2 Horizontal or Vertical for Certain

T

–

2020

v1

EllipseInscribed Triangles VI: Envelope of X4 Loci is AreaInvariant and Cousin of Pascal’s Limaçon

T

–

2020

v1

Inversive N=3
Title

sound

Kimberling Centers

Year

urls

Circles Galore I: Loci of FocusInversive 3Periodics in the Elliptic Billiard (11 notable centers)

T

1, 2, 3, 4, 5, 9, 10, 11, 20, 40, 100

2020

v1

Circles Galore II: 29 Loci of FocusInversive 3Periodics in the Elliptic Billiard

T

1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 20, 21, 35, 36, 40, 46, 55, 56, 57, 63, 65, 72, 78, 79, 80, 84, 90, 100

2020

v1

Circles Galore III: Loci of FocusInversive 3Periodics in the Elliptic Billiard (9 notable centers)

T

1,2,3,4,5,8,9,10,11

2020

v1

Locus App
Title

sound

Kimberling Centers

Year

urls

Loci of EllipseInscribed Triangles: Part 01  Intro to the App

T

–

2020

v1

Loci of EllipseInscribed Triangles: Part 02  The Homothetic Family

T

–

2020

v1

Loci of EllipseInscribed Triangles: Part 03  Derived Triangles

T

–

2020

v1

Loci of EllipseInscribed Triangles: Part 04  Locus Type

T

–

2020

v1

Inverse Curvature
Title

sound

Kimberling Centers

Year

urls

Algebraic Miracle: The ZeroArea N=3 Homothetic Poncelet InverseCurvature Polygon (ICP)

T

–

2020

v1

Hyperbolic Billiard
Title

sound

Kimberling Centers

Year

urls

3Periodics in a Hyperbolic Billiard

T

–

2020

v1

Videos N≥3
Early Results
Title

sound

N

Year

urls

Stationary Circle for N=5

T

5

2019

v1

Stationary Circles


3 to 8

2019

v1

Generalization of the Stationary Mittenpunkt and CausticSweeping Extouchpoints


4,5

2019

v1

Inconics
Title

sound

N

Year

urls

Excentral X(3)Centered & MacBeath Inconics: Invariant Aspect Ratio

T

NULL

2020

v1

N>3 Periodics
Title

sound

N

Year

urls

4periodics and Monge’s Orthoptic Circle


4

2019

v1, v2

4periodics: Loci of Triangle Centers for Vertex Triad


4

2019

v1

5periodics: locus of P1,P2,P3 triangle


5

2019

v1

5periodics: locus of P1,P2,P4 triangle


5

2019

v1

5periodics: Loci of Subtriangles (123 and 124)


5

2019

v1

Upright 5periodic family


5

2019

v1

6periodic family


6

2019

v1

Octagramma Mysticum


8

2019

v1, v2

Enagramma Mysticum: loci of side intersections


9

2019

v1

Mittenpunktlike Construction of a Stationary Point


4 to 7

2019

v1

Generalized Mittenpunkt and OnCaustic Extouchpoints


4,5

2019

v1

Family of Orbits and Their Caustics


3 to 6

2019

v1

EllipseInscribed Parallelogram: invariants of the Pedal Polygon w/ respect to boundary points

T

4

2020

v1, v2

Elliptic Billiard with Perpendicular Reflection Rule

T

n/a

2020

v1

Invariant Area Ratios to MinimumArea Steiner Pedal Polygons

T

5

2020

v1

Circumcircles of Focus with Consecutive Vertices Homothetic to Focus Antipedal

T

5,6

2020

v1

Incenters of Focus Triads: Invariant Area Ratio to NPeriodic and Elliptic Locus

T

5

2020

v1

An Invariant Based on Inradii and Circumradii of Subtriangles in the Elliptic Billiard

T

5

2020

v1

SelfIntersected
Title

sound

N

Year

urls

Selfintersecting 4periodics (bowtie and tangential polygon)


4

2020

v1, v2

Selfintersecting 5periodics (pentagram)


5

2019

v1

Selfintersecting 5periodics (pentagram): Locus of Internal Intersections


5

2019

v1

Elliptic Billiard: SelfIntersected 6Periodics (type I)

T

6

2020

v1

Elliptic Billiard: SelfIntersected 6Periodics (type II)

T

6

2020

v1

SelfIntersected 6Periodics in the Elliptic Billiard: Invariant Perimeter FocusInversive Polygon

T

6

2020

v1

Elliptic Billiard 8Periodics: Null sum of double cosines of outer polygon


8

2020

v1

Family of SelfIntersecting 4Periodics in the Elliptic Billiard: Inversive Polygon is a Segment

T

4

2020

v1

Type II SelfIntersected 8Periodics in the Elliptic Billiard + Outer & Inversive Polygons

T

8

2020

v1

Type I SelfIntersected 8Periodics in the Elliptic Billiard and the Inversive Polygon

T

8

2020

v1

Elliptic Billiard SelfIntersected 7Periodics, a/b=2: Invariant Perimeter FocusInversive Polygons

T

7

2020

v1

Family of selfintersected N=8 w/ turning number 2 in the Elliptic Billiiard

T

8

2020

v1

The two types of selfintersected 7periodics in the Elliptic Billiard

T

7

2020

v1

Elliptic Billiard: Vertices of SelfIntersected 4Periodics & Outer Polygon are concyclic w/ foci

T

4

2020

v1

Tangential Polygon
Title

sound

N

Year

urls

Locus of Vertices of the Excentral Polygon


3 to 6

2019

v1

5periodics and feet of excenters


5

2019

v1

Locus of meetpoints of ExcentraltoOrbit Perpendiculars


3,4,5

2019

v1, v2, v3

Pencil of Confocals
Title

sound

N

Year

urls

Tangents from a point on boundary to caustics


–

2019

v1

Tangents to caustics from billiard’s vertex lie on a single circle


–

2019

v1

Loci of tangents to confocals: point traverses entire elliptic boundary


–

2019

v1

Loci of tangents to confocals: point traverses neighborhood of right vertex


–

2019

v1

Locus of tangents from ellipse: 45,45 degrees starting points


–

2019

v1

Locus of tangents from ellipse: 5,95,45,45 degrees starting points


–

2019

v1

Stationary Circles
Title

sound

N

Year

urls

5periodics and a stationary circle


5

2019

v1

Stationary circles for N=3 to 8


3 to 8

2019

v1

Misc
Title

sound

N

Year

urls

Elliptic Billiards in Brazil

T

4

2019

v1

Reuleaux Triangle: Properties of Negative Pedal Curve, and Exploring its Billiard Trajectories

T

n/a

2020

v1

NonConcentric Circular Poncelet Pair: Invariant Sum of Japanese Theorem Inradii (A. Akopyan)

T

5,6

2020

v1

Pascal’s Limaçon as Envelope of Circles

T

–

2020

v1

Pedal Invariants
Title

sound

N

Year

urls

Concyclic Feet of Focal Pedals and Invariant Product of Sums of Lengths for odd N

T

5,6

2020

v1

Invariant sum of squared altitudes from each focus to tangential polygon sides

T

3,4,5,6,7,8

2020

v1

Altitude Invariants to NPeriodics and their Tangential Polygons (N=3,4)

T

3,4

2020

v1

Altitude Invariants to NPeriodics and their Tangential Polygons (N=5,6)

T

5,6

2020

v1

Sum of square altitudes from arbitrary point to Nperiodic tangents is invariant

T

5

2020

v1

Pedal polygons from each focus have invariant area product

T

5

2020

v1

Pedal Polygons for the NPeriodic and its Tangent Polygon: Area Ratio Invariances

T

5,6

2020

v1

Exploring Amazing Invariants of NPeriodics and their Pedal Polygons

T

3–12

2020

v1

Centroid Stationarity (even N)

T

4,6

2020

v1

Steiner’s Hat
Title

sound

N

Year

urls

Equal sum of distances from each focus to vertices of antipedal polygon

T

3,4,5,6

2020

v1

Pedal Polygons to NPeriodics with respect to a Focus: Concyclic Vertices and Circular Caustic

T

3,4,5,6

2020

v1

The Envelope of Ellipse Antipedals is a ConstantArea Deltoid

T

n/a

2020

v1, v2

A narrated tour of the Garcia Deltoid: Surprising Invariants and Properties

T

n/a

2020

v1

Properties of Osculating Circles to the Ellipse at the 3 Cusp PreImages

T

n/a

2020

v1

Locus of Cusps and Deltoid Center of Area

T

n/a

2020

v1

Concyclic preimages, osculating circles, and 3 areainvariant triangles

T

n/a

2020

v1

Rotated Negative Pedal Curve of Ellipse is AreaInvariant

T

n/a

2020

v1

Area Invariants
Title

sound

N

Year

urls

Amazing Ellipse Pedal and Contrapedal Curves: area invariance for all pedal points on a circle!

T

n/a

2020

v1

Regular Polygons: the Signed Area of the Antipedal Polygon Vanishes along a Circle?

T

3,4,5,6,7,8

2020

v1

Steiner’s KrümmungsSchwerpunkt implies AreaInvariant Interpolated Pedal Curve over Circles

T

–

2020

v1

Exotic Billiards
Title

sound

N

Year

urls

HorizontalVertical Billiard in a Rhombus and Parallelogram: are there NPeriodics?
