17-Two area triangle theorem

Let  be a triangle. Let a line  on the plane. Construct three parallel line through  respectively meet  at .  meets  at  

if  we difine  else . Difine sign, sign cyclically.

First Area theorem:

sign(A1).area(△A1BC)+sign(B1).area(△B1AC)+sign(C1).area(△C1AB)=2.area(△ABC)

Secon Area theorem:

∥sign(A1).area(△AB1C1)+sign(B1).area(△BA1C1)+sign(C1).area(△CA1B1)∥=area(△ABC)