If four points $A (¯ a), B (¯ b), C (¯ c)$ and $D (¯ d)$ are coplanar then show that $[¯ a ¯ b ¯ d] + [¯ b ¯ c ¯ d] + [¯ c ¯ a ¯ d] = [¯ a ¯ b ¯ c]$

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Solution

Given
$[\to a \to b \to d] + [\to b \to c \to d] + [\to c \to a \to d]$

$= [\to a \to b λ \to c] + [\to b \to c λ \to a] + [\to c \to a λ \to b]$ ..... [Since $A, B, C, D$ are coplanar]
$= λ [\to a \to b \to c] + λ [\to c \to b \to a] + λ [\to a \to b \to c]$
$= 3 λ [\to a \to b \to c]$
$= [\to a \to b \to c]$

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