Quiz

In the diagram, the circumference of the external large circle is
1) longer, or
2) shorter, or
3) equal to,
the sum of the circumferences of all inner circles centered on the common diameter, tangent to each other.

New Geometry 新几何

New Geometry (新几何) invented by Zhang JingZhong (張景中) derived from 2 basic theorems:

1) Triangles internal angles =180º

2) Triangle Area = ½ base * height
=> derive all geometry
=> trigonometry
=> algebra
(These 3 maths are linked, unlike current syllabus taught separately)

The powerful Area (Δ) Proof Techniques:

1) Common Height:
Line AMB, P outside line
Δ PAM / Δ PBM = AM/BM

2) Common 1 Side (PQ):
Lines AB and PQ meet at M
Δ APQ /Δ BPQ = AM/BM

3) Common 1 Angle:
∠ABC=∠XYZ (or ∠ABC+∠XYZ = ∏ )
Δ ABC /Δ XYZ= AB.BC /XY.YZ

These 3 theorems can prove Butterfly and tough IMO problems.

Axiom

Axiom (Greek): meant request. The reader is requested to accept the axioms unquestioningly as the rules of the game.

Euclid's "Element" built the whole Geometry with only 5 axioms.
The 5th axiom "Parallel line" was not challenged for 3,000 years until 19th CE Gauss & Riemann developed the Non-Euclidian Geometry.

Klein's Geometry in Group

This is the "New Geometry" introduced by Klein 200 years ago in his Erlangan Program (his PhD Thesis).

Rigid Motion is defined by 3 components: Translation, Rotation and Reflection.

If fixed at one point (origin), there is no translation, only Rotation ρ(θ) and Reflection r(θ) are possible around that fixed point.

We can prove r(θ) and ρ(θ) form a Group O2, namely Orthogonal Group with this property:

$latex A^{T}. A = A. A^{T} = I $

where A can be any of the 2 matrices represented by ρ(θ)

or r(θ),

$latex A^{T} $ is the transpose of A (columns => rows, rows => columns).

1. Rotation

ρ(θ)=

(cos θ  -sin θ)

(sin θ   cos θ)

2. Reflection

r(θ) =

(cos θ   sin θ)

(sin θ   -cos θ)

when θ =0,

r0 =

(1 0)

(0 -1)

=> r(θ) = ρ(θ).r0

Change of Reference Axis:

 Make a shift from fixed origin A to another fixed original A' by a translation t(α), the first Orthogonal Group O at A and the second Orthogonal Group O' at A' are related by:

O' = t(α).O.t^-1(α)
ρ'(θ) = t(α).ρ(θ).t^-1(α)
r'(θ) = t(α).r(θ).t^-1(α)

Note: this looks analogous to Conjugate groups (Normal Subgroups).

Einstein Relativity interpreted by Rigid Motion (M4).

If first origin A is the Earth, second origin A' is the spaceship traveling at speed of light, ie t(α) = c

O' = t(α).O.t^-1(α) ; O & O' ∈ O4

<=> O'.t(α) = t(α).O

Vector Algebra

Vector changes Geometry to Algebra

1. No complexity of Analytical Geometry
2. Remove the astute dotted (helping) line in Geometry
3. No need diagram: Use only 2 vector properties:
Head- to-Tail:
$latex \vec{AC}=\vec{AB}+\vec {BC}$
Closed Loop:
$latex \vec{DE}+\vec{EF}+\vec{FD}=0$
4. Enable Computer automated proof of Geometry via Algebra.

Example: 任意四边形 Quadrilateral ABCD with M,N midpoints of AB, CD, resp.
Prove： MN=1/2(BC+AD)
Proof: (by vector):

Consider MBCN:
MN=MB+ BC+ CN..(1)

Consider MADN:
MN=MA+ AD+ DN..(2)

(1) +(2):
2MN=(MB +MA) +
(BC +AD) +(CN +DN)

but (MB +MA) =0,
(CN +DN) =0 [same magnitude but different direction cancelled out ]

=> MN=1/2 (BC +AD)

Special cases:
1. A = B (=M)
=> triangle ACD
AN = 1/2 (AC +AD)
2. BC // AD
=> Trapezium ABCD
MN=1/2 (BC +AD)
=> MN // BC // AD

Automorphism = Symmetry

Automorphism of a Set is an expression of its SYMMETRY.
1. Geometry figure (e.g. triangle) under certain transformations (reflection, rotation, ...), it is mapped upon itself, certain properties (distance, angle, relative location) are preserved.
=> the figure admits certain automorphism relative to its properties.
2. Automorphism of an arbitrary Set (with arbitrary relations between its elements) form an Automorphism Group of the set.