Given: 3 circles of radius 2

Find: Shaded Area?

Hint:

Sum of 3 angles in the triangle

= **π **= 1/2 (2**π) ＝1/2 circle **

**=> its Area = 1**/2(**π. 2^2) = 2π = Sum of 3 arcs’ area**

Euclid

Given: 3 circles of radius 2

Find: Shaded Area?

Hint:

Sum of 3 angles in the triangle

= **π **= 1/2 (2**π) ＝1/2 circle **

**=> its Area = 1**/2(**π. 2^2) = 2π = Sum of 3 arcs’ area**

**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):**

http://m.dangdang.com/touch/product.php?pid=21120565&sid=bb1368df71a7809d&recoref=search

References:

https://tomcircle.wordpress.com/2013/04/21/new-geometry-%e6%96%b0%e5%87%a0%e4%bd%95/

**The famous Butterfly Geometry Problem**:

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’ !

http://news.yahoo.com/vatican-asked-honour-jesuit-tried-evangelise-china-193740033.html

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

Circumference of the external circle

= sum of circumferences of all inner circles

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.