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 SM1 is the median of ?PQS.
The median passes through vertex S and bisects the opposite side PQ.
(3) Symmedians: When the median SM1 is reflected on the internal angle bisector SS' another line is formed called the symmedian, denoted SSS 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 ?PQS.
(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 ?PQS so that:
SP+PSSP=SSPQ+QS
Points and Centers
(1) Incenter-Ci. 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- CE. 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-CC. 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- CO
The three altitudes concur at the point called the orthocenter.
Co = SF1?PF2?QF3
(5) Centroid- Ct
The three medians concur at the point called the centroid- Ct.
Ct = SM1?PM2?QM3
(6) Nine-Point Center- C9.
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 ?PQS, 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 ?PQS, 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 ?PQS concur at a point called the Nagel point. It is also the point of concurrence of the three splitters of ?PQS. 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 M1, M2 and M3 are the respective midpoints of the sides of ?PQS.
(13) Mittenpunkt Point (middlespoint): This is the perspector of the medial triangle M1M2M3 and the excentral triangle CE1CE2CE3. Also, the three symmedians of the excentral triangle concur in this point. The trilinear coordinates follow.
CMt=?2?-Sai2e1,e2,e3 =2?2P2?-Sai2cotß2,cot02,cot?2 =?2Sai2-P22a1-P,2a2-P,a3-P
The following points are also discussed later in the text.
M13 = M1CC n PS
M12 = M1CC n QS
L1 = SCC n PQ
M23 = M2CC n PS
M21 = M2CC n PQ
M32 = M3CC n SQ
M31 = M3CC n 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 ? PQS, is the triangle formed by the intersections of the three lines joining the vertices of ? PQS to T and the sides of ? PQS, as depicted below.
Triangle PQS may be referred to as the anti-cevian triangle of triangle PCQCSC.
Let T=U,V,W?Area ?PCQCSC=a1a2a3?UVWa1U+a2Ua2V+a3Wa3W+a1U
(2) Pedal Triangle The Pedal triangle PPQPSP of a Point T relative to ?PQS, is the triangle formed by the perpendicular projections of T onto the side lines, as shown below.
Let T=U,V,W?Area ?PPQPSP=a1,a2,a3·VW,WU,UV?2a1a2a3=rC2-CCT2?8rC2
(3) Perspective Triangles
Two triangles
