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.