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.