Modern Algebra for Ancient Geometry: Volumes One & Two

Preface

My first Rubik’s cube was the beginning of ongoing addiction with solving puzzles. The first time I won a game of chess against a worthy opponent served as the introduction to a magic (and sometimes tragic) world of frustration and ecstasy. The first time I soloed over a series of chord changes made me a slave to improvised music. The first time I understood how to apply the Convolution Theorem to resolve a probability question- the list goes on and on! To fuel my passions, I became a professional mathematician, performing actuarial calculations, analysing population dynamics, and taming a variety of chaotic systems while lecturing and tutoring in pure and applied mathematics.

In my spare time I began to pursue other interests such as woodwork, lutherie (stringed instrument construction), making puzzles out of rare timbers, as well as architecture, building and renovating houses. The fact that I was an academic laying bricks, or a brickie who happened to also give lectures on Stochastic Dynamics, enabled me to appreciate the fractal dimensions of both these very different worlds. My wife and I designed and built our home, and on the day when the glass windowpanes had to be cut to size, one opening required a circular segment with six equal panes, a subset of a regular 25-agon. A simple little problem of geometry? Not so! The mortar that binds the building blocks of higher mathematics is its notation, which is the part that’s often feared and frequently done incorrectly, causing the collapse of the metaphorical (and sometimes the physical) building! Thankfully our house suffered no such fate, but the experience led me to the idea of formulating simple geometric problems so they could easily be used by those seeking their solution.

As time passed, I began to study the beautiful geometries of triangles and the various points within them. The way the lines interplayed with various circles and conic sections was life changing. The further I dived, the more excited I became. I also found myself perturbed with why this is not better known, and why is it not taught at school in tandem with the centuries old Euclidian geometry. I began to ask valued persons, knowledgeable in mathematics and otherwise, whether they’d heard of such things as the Symmedian Point, the Brocard Circle, the Morley Triangle, Lemoine’s Theorem, Steiner Chains, Yff Parabola, Keipert’s Hyperbola, the Nagel Line; each time their answer was invariably no. I wasn’t surprised!

Anyone today who wishes to pursue modern geometry (whatever that is) does not need to be familiar with the synthetic geometric treatment of conic sections as done by the ancient Greeks, whilst many professions require no geometry to be known at all. However, in my opinion, life can become more meaningful and enjoyable when one knows how to locate the centroid of a given triangle or determine the area of a triangle with the sides known (non-right triangles) without using trigonometry.

The purpose of this book is to attract the curious. The ideas shared here are fundamentally simple, and the mathematics is optional. The fun is in enjoying the creativity without fearing the formulas. Throughout these pages lurk points, lines, triangles, circles, and conics that are so valid and important that they’ve been named! Their personality is described while their DNA is written in algebra.

Artists of all mediums can enjoy the beautifully properties of the triangle and its many offshoots when creating their own logos and works of art. Those versed in elementary algebraic techniques can have fun discovering elusive geometric proofs and inventing their own. Those who wish to become better conductors of their own lives are encouraged to read on. When was the last time you did something for the first time?

There are two tragedies in life; when you get what you want and when you don’t; the rest is the groundwork for their occurrence!

B.T.

I am not a mathematician; I’m a doctor who likes doing sums. During my exploration of the world of mathematics, I have found a few gems and produced a great deal of rubbish. Bill has always been supportive, and therefore when he told me about this book, I offered to help, little expecting it would take over three years.

At first, I had some doubts! What is the point? Who will read it? Now, after thousands of hours of work, I will try to answer these questions.

What is the point? There is no practical use for a Renoir painting, its existence is its own justification; it needs no other. Within these pages are some amazing theorems and beautiful mathematics. Just as for great art, there is no need to justify this work, the mathematics is its own justification. I humbly compare this book to great art because it is a collection of the works of great men; Archimedes, Apollonius, Brocard, Euler, Tucker, Yff, Morley, and many others, and I hope it will be seen as such.

Who will read it? Among the billions of earthlings, there must be a few slightly mad individuals like Bill and me, at least I hope so.

Read on, absorb some of it, study some of it, and try to enjoy most of it.

E.W.

Volume One

Chapter 1

Metrics

What is the currency with which to purchase the purest understanding of both descriptive and analytic geometry? The answer is an algebra that is self-sufficient and well defined to aid us in the manoeuvres of pure and applied scenarios.

This book aims for this very purpose and lays out the foundation for such algebra. The usual field and vector operations are intact whilst several new concepts are defined for the first time.

The many metric and trigonometric theorems that follow are to be harvested for use in complicated proofs, and by no means are meant to be committed to memory. The examples that are included will enable the reader to recognise the simplicity of the design yet appreciate the potential of the method.

Two co-ordinate systems have finally been combined. The rarely used Trilinear Co-ordinate system can be transformed to the better-known Cartesian system and vice versa. What seems impossible in one system may become possible in the other. It is now our choice!

The following diagram illustrates the difference in notation between this book and that used in Kimberling’s Encyclopaedia of Triangle Centers, which is referenced extensively throughout this text. The Kimberling notation is in red.

As a preliminary step, we define the following terms which are extensively used in the following text.

The reference triangle

An arbitrary triangle-PQS, with its lengths and angles as shown below.

(1) Angle Bisectors

(i) Internal bisector

(ii) External bisectors

(2) Medians: The line SM 1 is the median of ΔPQS.

The median passes through vertex S and bisects the opposite side PQ.

(3) Symmedians: When the median SM 1 is reflected on the internal angle bisector SS' another line is formed called the symmedian, denoted SS S in the diagram below.

SM1 and SSS are said to be isogonal conjugates of one another.

(4) Perpendicular Bisectors

(5) Altitudes: A line passing through vertex (S) perpendicular to the opposite edge PQ is an altitude of ΔPQS.

(6) Cleavers: A perimeter bisecting line through the mid-point of a side is a cleaver.

The diagram above shows the cleaver M1SCL such that:

SCLP→+PM1→=M1Q→+QS→+SCLS→

(7) Splitters: A perimeter bisecting line issuing from a vertex is a splitter.

The splitter SSP bisects the perimeter of ΔPQS so that:

SP→+PSSP→=SSPQ→+QS→

Points and Centers

(1) Incenter-C i . The three internal angle bisectors concur at a point called the Incenter.

This point is the center of the incircle. i.e. Ci = SS′⋂PP′⋂QQ′

(2) Excenter- C E . Two adjacent external angle bisectors, and the opposite internal angle bisector concur at a point called the excenter. There are three such centers associated with any triangle. Their locations and other measures will be considered later in the text.

Example: Excenter opposite to S

(3) Circumcenter-C C . The three perpendicular side bisectors concur at the point called the circumcenter.

This is the center of the circle that passes through the three vertices P, Q and S. The circumcircle is referred to as the fundamental circle (canonical) for the triangle geometry that follows.

(4) Orthocenter- C O

The three altitudes concur at the point called the orthocenter.

Co = SF1⋂PF2⋂QF3

(5) Centroid- C t

The three medians concur at the point called the centroid- Ct.

Ct = SM1⋂PM2⋂QM3

(6) Nine-Point Center- C 9 .

This point is the center of a special circle that passes through the three mid-points of the sides, the three feet of the altitudes and the three Euler points (mid points from the vertices to the orthocenter) of ΔPQS, amongst many others. It is treated in depth later in the text.

(7) Symmedian Point

The three symmedians concur at the symmedian point; also referred to as the Lemoine point.

(8) Gergonne Point

The three line segments joining each vertex to the opposite tangency point between the incircle and ΔPQS, concur at a point called the Gergonne point.

CG = ST1⋂PT2⋂QT3

(9) Nagel Point

The three line segments joining a vertex to the tangency point between the opposite excircle and ΔPQS concur at a point called the Nagel point. It is also the point of concurrence of the three splitters of ΔPQS. See ex-touch triangle later in the text.

(10) Feuerbach Point

The tangency point between the nine-point circle and the incircle is called the Feuerbach point denoted-Ji.

(11) Harmonic conjugate of the Feuerbach point

The three line segments joining each vertex to the opposite tangency point between the nine- point circle and the excircles are concurrent at a point called the harmonic conjugate of the Feuerbach point, denoted- JH.

(12) Cleavance Center: The three cleavers concur at a point called the Cleavance Center, also referred to as the Spieker center. The points M 1 , M 2 and M 3 are the respective midpoints of the sides of ΔPQS .

(13) Mittenpunkt Point (middlespoint): This is the perspector of the medial triangle M 1 M 2 M 3 and the excentral triangle C E 1 C E 2 C E 3 . Also, the three symmedians of the excentral triangle concur in this point. The trilinear coordinates follow.

CMt=λ2η−∑αi2ε1,ε2,ε3          =2λ2P2η−∑αi2cotβ2,cot02,cotγ2         =λ2∑αi2−P22α1−P,2α2−P,α3−P

The following points are also discussed later in the text.

M13 = M1CC ∩ PS

M12 = M1CC ∩ QS

L1 = SCC ∩ PQ

M23 = M2CC ∩ PS

M21 = M2CC ∩ PQ

M32 = M3CC ∩ SQ

M31 = M3CC ∩ PQ

There are many other interesting points, and the reader is encouraged to explore Kimberling’s Encyclopaedia of Triangle Centers, which presents a comprehensive list.

General Triangles

When comparing two triangles, the following definitions apply.

(1) Cevian Triangle The cevian triangle of a point T, relative to Δ PQS , is the triangle formed by the intersections of the three lines joining the vertices of Δ PQS to T and the sides of Δ PQS , as depicted below.

Triangle PQS may be referred to as the anti-cevian triangle of triangle PCQCSC.

Let   T=U,V,W⇒Area ΔPCQCSC=α1α2α3λUVWα1U+α2Uα2V+α3Wα3W+α1U

(2) Pedal Triangle The Pedal triangle P P Q P S P of a Point T relative to ΔPQS , is the triangle formed by the perpendicular projections of T onto the side lines, as shown below.

Let  T=U,V,W⇒Area ΔPPQPSP=α1,α2,α3⋅VW,WU,UVλ2α1α2α3=rC2−CCT→2λ8rC2

(3) Perspective Triangles

Two triangles 𝑃𝑄𝑆 and 𝑃′𝑄′𝑆′ are said to be in perspective from O, if their three pairs of corresponding vertices are joined by lines which meet in O, as shown below.

A collection of interesting Triangles

(1) Medial Triangle

The mid- points of the sides of ΔPQS are the vertexes of the medial triangle.

(2) Orthic Triangle: This triangle has its vertices on the feet of the altitudes of Δ 𝑃𝑄𝑆 , as shown below.

Triangle 𝐹1𝐹2𝐹3 is both the Cevian triangle and the Pedal triangle with respect to the orthocenter of ∆𝑃𝑄𝑆.

(3) Euler Triangle: The vertices of this triangle are G 1 , G 2 and G 3, the mid-points of the lines joining the orthocenter to the vertices of ΔPQS

Note: ΔG1G2G3 is the Euler triangle, ΔF1F2F3 is the Orthic triangle and ΔM1M2M3 is the Medial triangle, and as all these points lie on the nine-point or Feuerbach circle, the nine-point center-C9 acts as the circumcenter of these three triangles. The many wonderful properties shared by these points, lines, triangles, and circles will be uncovered using analytic methods later in the text.

(4) Incentral Triangle

The vertices of the Incentral Triangle are formed by the intersection of the angle bisectors with the opposite sides.

(5) Excentral Triangle

This triangle has vertices at the center of the escribed circles of 𝛥𝑃𝑄𝑆 as depicted below.

(6) Intouch Triangle: This triangle is also called the contact triangle. Its vertices are formed by the tangency points of the incircle and the sides of 𝛥𝑃𝑄𝑆 as depicted below.

Thus, ΔT1T2T3 is the Pedal triangle of ΔPQS with respect to the incenter-Ci, and the Cevian triangle with respect to the Gergonne point-CG.

(7) Extouch Triangle: The triangle having its vertices on the tangency points between 𝛥𝑃𝑄𝑆 and the three excircles is called the extouch triangle. It is denoted 𝛥𝑈 1 𝑉 2 𝑊 3 in the diagram below.

Thus, 𝛥𝑈1𝑉2𝑊3 becomes the Cevian triangle with respect to the Nagel point- 𝐶𝑁, and the Pedal triangle with respect to the Bevan point- 𝐶𝐵𝑉.

The above diagram shows the original Δ𝑃𝑄𝑆 together with its Excentral 𝛥𝐶𝐸1𝐶𝐸2𝐶𝐸3 and its Extouch 𝛥𝑈1𝑉2𝑊3. Nagel’s point- 𝐶𝑁 and the Bevan point- 𝐶𝐵𝑉 are also highlighted.

(8) Intangents Triangle: The three edges of this triangle are tangential to the incircle and internally tangential to the excircles of ΔPQS as shown below.

(9) Extangents Triangle: The three edges of this triangle act as external tangents to the excircles of 𝛥𝑃𝑄𝑆. This triangle and ∆𝑃∆𝑄𝑆 are in perspective form the orthocenter- 𝐶 𝑂 .

(10) Tangential triangle: The tangential triangle is formed by the tangents to the circumcircle at the vertices 𝑃, 𝑄 and 𝑆 -labelled as 𝑇 𝑃 𝑇 𝑄 𝑇 𝑆 in the diagram below.

(11) Symmedial Triangle .

The vertices of this triangle are the points of intersection of the symmedians with the sides of ΔPQS, as shown in the following diagram.

The symmedial triangle is clearly the Cevian triangle of ΔPQS with respect to the symmedian point-Csy. The diagram below shows the relationships of the Centroid-𝐶𝑡, the Incenter-𝐶𝑖 and the Symmedian Point-𝐶𝑠𝑦.

(12) Anticomplementary Triangle

This triangle has the reference 𝛥𝑃𝑄𝑆 as its medial triangle.

The diagram above shows the Anticomplementary triangle 𝛥𝐴𝑆𝐴𝑃𝐴𝑄, the reference 𝛥𝑃𝑄𝑆 and its medial 𝛥𝑀1𝑀2𝑀3. All three are in perspective from 𝐶𝑡, the centroid of 𝛥𝑃𝑄𝑆, as well as being Cevian triangles, from largest to smallest, with respect to the same point 𝐶𝑡.

(13) Mid-Arc Triangle

This triangle’s vertices co inside with the nearest points of intersection of the internal angle bisectors and the incircle of 𝛥𝑃𝑄𝑆.

The circumcircle of the Mid-Arc 𝛥𝑃𝑖𝑄𝑖𝑆𝑖 is the incircle of 𝛥𝑃𝑄𝑆, as shown above.

(14) Mid- Altitude Triangle: This triangle has its vertices on the mid-points of the altitudes of ΔPQS, as depicted below.

The vertices PA, QA and SA are such that:

𝑃𝐴 = 𝑀𝑖𝑑 𝑃𝑡 𝑜𝑓 𝑃𝐹2        𝑄𝐴 = 𝑀𝑖𝑑 𝑃𝑡 𝑜𝑓 𝑄𝐹3          𝑆𝐴 = 𝑀𝑖𝑑 𝑃𝑡 𝑜𝑓 𝑆𝐹1

(15) Hexyl Triangle: This triangle’s vertices are 𝑃 𝐻 , 𝑄 𝐻 𝑎𝑛𝑑 𝑆 𝐻 such that:

𝑃𝐻 = 𝐶𝐸3𝑊1 ∩ 𝐶𝐸1𝑈3       𝑄𝐻 = 𝐶𝐸1𝑈2 ∩ 𝐶𝐸2𝑉1        𝑆𝐻 = 𝐶𝐸2𝑉3 ∩ 𝐶𝐸3𝑊2

Note: The vertices, combined with the centers of the ex-circles, form a hexagon.

(16) Circumcircle Mid-Arc Triangle

The vertices of this triangle are the circumcircle mid-arc points of ΔPQS, denoted by M¯1M¯2M¯3 in the following diagram.

(17) Circum-Orthic Triangle

This triangle is the Circum-Cevian triangle of ΔPQS with respect to the orthocenter-CO, denoted by POQOSO in the following diagram.

(18) Circum-Medial Triangle

This is the Circum-Cevian triangle of ΔPQS with respect to its centroid-Ct, labelled PMQMSM.

(19) Inner Napoleon Triangle

The triangle formed by the centers of the equilateral triangles, erected internally on the sides of the reference ΔPQS, is called the inner Napoleon triangle. According to Napoleon’s theorem, it is also equilateral. The proof is provided later in the text.

(20) Outer Napoleon Triangle

The triangle formed by the centers of equilateral triangles, erected externally on the sides of the reference ΔPQS, is called the outer Napoleon triangle. It is also an equilateral triangle. The proof follows later in the text.

(21) First Circum- Brocard Triangle

The line segments 𝑃𝐶𝐵, 𝑄𝐶𝐵 𝑎𝑛𝑑 𝑆𝐶𝐵 extended to the circumcircle of 𝛥𝑃𝑄𝑆, form the vertices of a triangle called the first Circum-Brocard triangle, labelled 𝐵𝑃𝐵𝑄𝐵𝑆 in the following diagram.

(22) Second Circum-Brocard Triangle

The line segments 𝑃𝐶′𝐵, 𝑄𝐶′𝐵 and 𝑆𝐶′𝐵 extended to the circumcircle of 𝛥𝑃𝑄𝑆 constitute the vertices of a triangle called the second Circum-Brocard triangle, labelled 𝐵′𝑃𝐵′𝑄𝐵′𝑆 in the following diagram.

ΔPQS is congruent to Δ𝐵′𝑃𝐵′𝑄𝐵′𝑆

(23) Morley Triangle

The three points of intersection of the adjacent angle trisectors form an equilateral triangle.

(24) Feuerbach Triangle

This triangle has its vertices at the tangency points between the nine-point circle and the three ex-circles.

(25) Inner Vecten Triangle

The triangle formed by the centers of the internally erected squares on the sides of reference 𝛥𝑃𝑄𝑆 is called the Inner Vecten triangle.

(26) Outer Vecten Triangle The triangle formed by the centers of the externally erected squares on the sides of reference 𝛥𝑃𝑄𝑆 is called the Outer Vecten triangle.

(27) Apollonius Triangle The vertices of this triangle are the points of tangency between the three excircles and the grand Apollonius circle of ∆𝑃𝑄𝑆 .

Metrics

In order to merge the descriptive and analytic branches of geometry, we need to establish a concise and simple relationship between trigonometric and basic metric quantities.

(1) Linear measures:

(𝑖) Perimeter = P = α 1 + α 2 + α 3 = ∑ i = 1 3 α i

(𝑖𝑖) The pseudo-perimeters are Є 1 , Є 2 𝑎𝑛𝑑 Є 3

Є1=α2+α3−α1 Є2=α3+α1−α2 Є3=α1+α2−α3

With g1=12Є1 g2=12Є2 g3=12Є3 g4=12P we have.

(2) Square measures

(𝑖) The area of  Δ P Q S = λ 2 = g 1 g 2 g 3 g 4   N o t e : P = ∑ i = 1 i = 4 g i

The quantity λ is a metric used throughout this work. Its many forms will be defined shortly.

(𝑖𝑖) δ 12 = 1 2 α 1 2 + α 2 2 − α 3 2 δ 13 = 1 2 α 1 2 + α 3 2 − α 2 2 δ 23 = 1 2 α 2 2 + α 3 2 − α 1 2

(𝑖𝑖𝑖) ∑ δ = δ 12 + δ 13 + δ 23 = α 1 2 + α 2 2 + α 3 2 2 = 1 2 ∑ α i 2

(𝑖𝑣) α 1 2 = δ 12 + δ 13 α 2 2 = δ 23 + δ 12 α 3 2 = δ 13 + δ 23

(𝑣) α 3 2 − α 2 2 = δ 13 − δ 12 α 2 2 − α 1 2