Triangle Signed Area Theorem

<p>For any oriented triangle ABC,

s(ABC) = s(AB) + s(BC) + s(CA)

Proof:

If OA ≡ a, OB ≡ b, OC ≡ c,

then

s(ABC) = 1/2 AB × AC

            = 1/2 (b - a) × (c - a)

            = 1/2[ b×c - b×a - a×c ]

            = 1/2[ a×b + b×c + c×a ]

Alternative form:

If A_1 ≡ [x_1, y_1], A_2 ≡ [x_2, y_2], A_3 ≡ [x_3, y_3] then

s(A_1 A_2 A_3) = (x_1y_2 - y_1x_2 + x_2y_3 - y_2x_3 + x_3y_1 - y_3x_1) / 2

Proof:

Immediate from Triangle Signed Area Theorem</p>