IMO Geometry Techniques 几何理论基础:分角定理、张角定理,推理证明

IMO Math usually contains 1 or 2 Geometry questions.

France, UK, Singapore, and some countries which reduce Secondary school syllabus in Euclidien Geometry, are disadvantaged in scoring Gold.


Geometry Proof : Sum of Squares


新几何 (张景中) New Geometry


The excellent and unique Geometry book is available at below link or 友谊书局 (新加坡 百胜楼 3楼 Bras Basah Complex, Victoria Street):



The famous Butterfly Geometry Problem:

Matteo Ricci – The 16th-Century Jesuit who brought Euclid to China


Matteo Ricci (利马窦) the Italian Jesuit Priest who came to China during the 16th Century  Ming Dynasty of WanLi 万历 Emperor, tried to evangelise the Imperial Court Mandarins with his impressive western Science knowledge and Mathematics. In1607 Ricci translated Euclid’s Masterpiece “The Elements” in Chinese with the Prime Minister Xu GuangQi (徐光啓), the later named Geometry as “几何”, a Chinese term still being used today. Paul Xu and few Scientist Officers were baptised by Ricci as the first Catholics in Chinese history. Ricci stopped the Euclid translation at Book 6 of 15 (Euclid’s 13 books + 2 add-on books from Ricci’s Jesuit professor Clavius), albeit the earnest plea from Xu to translate the remaining 9 books (likely because Ricci himself was not taught in Rome before he left for China), which were later translated by a British Alexander Wylie and a Chinese mathematician 李善兰 in 1857 of Qing Dynasty.

Ricci spent his remaining life in China because the Catholic Pope in Rome, influenced by the Macau-based Franciscans (then the opponent of Jesuits), prosecuted him for being ‘compromised’ by Chinese rituals of Ancestor Worship. 

Now the current Pope Francis wants to befriend China, which broke diplomatic tie with Vatican in 1950s, intents to beatify Matteo Ricci : The first Saint ‘Mathematician’ !



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.

Answer: 3) equal

circumference = π. diameter

Let d be the diameter of the external large circle C
Let dj be the diameter of the inner circle Cj

\displaystyle d = \sum_{j} d_j
\displaystyle \pi. d = \pi. \sum_{j} d_j= \sum_{j}\pi.d_j

Circumference of the external circle
= sum of circumferences of all inner circles

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

2) Common 1 Side (PQ):
Lines AB and PQ meet at M

3) Common 1 Angle:
∠ABC=∠XYZ (or ∠ABC+∠XYZ = ∏ )

These 3 theorems can prove Butterfly and tough IMO problems.


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:
A^{T}. A = A. A^{T} = I
where A can be any of the 2 matrices represented by ρ(θ)
or r(θ),
A^{T} is the transpose of A (columns => rows, rows => columns).
1. Rotation
(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:
\vec{AC}=\vec{AB}+\vec {BC}
Closed Loop:
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.