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.

1 Our Main Pages

  1. Invariants of 3- and N-Periodics in an Elliptic Billiard, with R. Garcia and J. Koiller.

  2. Loci of N-Periodics and their Derived and Internal Triangles

  3. new Envelopes and Evolutes: beautiful curves enveloped by pairs \(X_i,X_j\) of Triangle Centers.

  4. new Loci of Centers of Ellipse-Mounted 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.

  5. new Extremal-Area Pedal and Antipedal Triangles: a discussion on maximal and minimal pedals and antipedals to a triangle, with M. Helman.

2 Videos N=3

All videos are available as a single playlist.

Original 2011

Title sound Kimberling Centers Year urls
3-Periodic trajectories 2011 v1, v2, v3
Locus of incenter is elliptic for family of 3-periodics 1 2011 v1
Locus of the incircle touchpoints is a higher-order 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 Anti-Complement 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
3-Periodics 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 non-elliptic 2019 v1
Non-Elliptic 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 self-intersections 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 3-periodics in the Elliptic Billiard T 3,5,140 2020 v1
Circular loci for X(140) and X(547) over 3-periodics 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 3-Periodics 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 4-arc 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 Anti-Tangents is Stationary Circle 9 2019 v1
Intersections of Excentral Triangle and its Reflection is a Circle 9 2019 v1

Unrolled 3-Periodics

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 3-periodic Poncelet-Steiner 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 X3-Centered 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
X1-Centered Circumconic & X40-Centered (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 X5-Centered Circumconics are Invariant & Equal T 1,3,10,40,100 2020 v1
Invariant aspect ratios for the Circumbilliard and Excentral X6-Ctr Circumconic T 1,3,9,40,100 2020 v1
Side-by-Side View of Poristic and 3-Periodic 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
Non-monotonic X(88) and the X(1)-X(100) envelope 1,88,100 2020 v1
The Thomson Cubic of 3-periodics 1,2,3,4,6,9 2020 v1
Locus and elliptic envelope of excircle tangents’ hexagon (side touchpoints) T 2020 v1
Six-Point Conic passes through Sideline Tangents to Excircles 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 3-periodic 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 3-Periodic 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 3-Periodic P1 and reflected P2 is Elliptic T 2020 v1
The Bat-Envelope 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

Ellipse-Mounted

Title sound Kimberling Centers Year urls
Ellipse-Mounted Triangles: Elliptic locus of the Orthocenter X(4) and suprising area invariance! T 4 2020 v1
Circle-Mounted 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 3-Periodics Identical to Poristic Triangles. T 1,3 2020 v1
Between a Circle and a Concentric Ellipse: Poncelet 3-Periodics Identical to Poristic Excentrals. T 3,4,5 2020 v1
3-Periodics in a Concentric Homothetic Poncelet Pair: Circular Loci of four Triangle Centers T 2,13,14,15,16 2020 v1
3-Periodics in a Homothetic-Rotated Poncelet Pair: stationary orthocenter and loci of X107 and X122 T 4,107,122 2020 v1

3 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 Caustic-Sweeping 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
4-periodics and Monge’s Orthoptic Circle 4 2019 v1, v2
4-periodics: Loci of Triangle Centers for Vertex Triad 4 2019 v1
5-periodics: locus of P1,P2,P3 triangle 5 2019 v1
5-periodics: locus of P1,P2,P4 triangle 5 2019 v1
5-periodics: Loci of Subtriangles (123 and 124) 5 2019 v1
Upright 5-periodic family 5 2019 v1
6-periodic family 6 2019 v1
Octagramma Mysticum 8 2019 v1, v2
Enagramma Mysticum: loci of side intersections 9 2019 v1
Mittenpunkt-like Construction of a Stationary Point 4 to 7 2019 v1
Generalized Mittenpunkt and On-Caustic Extouchpoints 4,5 2019 v1
Family of Orbits and Their Caustics 3 to 6 2019 v1
Ellipse-Inscribed 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 Minimum-Area Steiner Pedal Polygons T 5 2020 v1

Self-Intersecting

Title sound N Year urls
Self-intersecting 4-periodics (bowtie and tangential polygon) 4 2020 v1, v2
Self-intersecting 5-periodics (pentagram) 5 2019 v1
Self-intersecting 5-periodics (pentagram): Locus of Internal Intersections 5 2019 v1

Tangential Polygon

Title sound N Year urls
Locus of Vertices of the Excentral Polygon 3 to 6 2019 v1
5-periodics and feet of excenters 5 2019 v1
Locus of meetpoints of Excentral-to-Orbit 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
5-periodics 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

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 N-Periodics and their Tangential Polygons (N=3,4) T 3,4 2020 v1
Altitude Invariants to N-Periodics and their Tangential Polygons (N=5,6) T 5,6 2020 v1
Sum of square altitudes from arbitrary point to N-periodic tangents is invariant T 5 2020 v1
Pedal polygons from each focus have invariant area product T 5 2020 v1
Pedal Polygons for the N-Periodic and its Tangent Polygon: Area Ratio Invariances T 5,6 2020 v1
Exploring Amazing Invariants of N-Periodics 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 N-Periodics with respect to a Focus: Concyclic Vertices and Circular Caustic T 3,4,5,6 2020 v1
The Envelope of Ellipse Antipedals is a Constant-Area 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 Pre-Images T n/a 2020 v1
Locus of Cusps and Deltoid Center of Area T n/a 2020 v1
Concyclic pre-images, osculating circles, and 3 area-invariant triangles T n/a 2020 v1
Rotated Negative Pedal Curve of Ellipse is Area-Invariant 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

Exotic Billiards

Title sound N Year urls
Horizontal-Vertical Billiard in a Rhombus and Parallelogram: are there N-Periodics? T n/a 2020 v1

Poncelet

Title sound N Year urls
Poncelet Family: Amazing Circular Locus of X3 and the Steiner’s Curvature Centroid T 5 2020 v1
5-Periodic Poncelet Families and their Pedal Polygons with Respect to their Curvature Centroids T 5 2020 v1
Jean-Victor Poncelet & Jakob Steiner walk into a Bierhaus + discover many invariants! Prost! Santé! T 3,4,5,6 2020 v1
N-Periodics on a Homothetic-Rotated Poncelet Pair: All Altitudes Meet at the Center! T 3,4,5,6,7 2020 v1
5- and 7-Periodics on a Homothetic-Rotated Poncelet Pair: All Altitudes Meet at the Center T 5,7 2020 v1

4 Interactive Applets

A p5.js interactive applet where you can easily inspect loci and other objects connected with the \(N=3\) family.