Prove that - 1 a b - a 1 c - b - c 1 - 1 - a2 - b2 - c2

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Properties of Determinants

Prove that , ...

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Prove that , $⎛ ⎜ ⎝ \begin{matrix} 1 & a & b - a & 1 & c - b & - c & 1 \end{matrix} ⎞ ⎟ ⎠ = 1 + a^{2} + b^{2} + c^{2}$

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∣ ∣ ∣ ∣ \begin{matrix} 1 & a & b - a & 1 & c - b & - c & 1 \end{matrix} ∣ ∣ ∣ ∣ =

a, b, c

a^{2} = b^{2} + c^{2} - 2 b c \sqrt{1 - a^{2}}, b^{2} = c^{2} + a^{2} - 2 c a \sqrt{1 - b^{2}}, c^{2} = a^{2} + b^{2} - 2 a b \sqrt{1 - c^{2}}

a = c \sqrt{1 - b^{2}} + b \sqrt{1 - c^{2}}

\frac{a sin (B - C)}{b^{2} - c^{2}} = \frac{b sin (C - A)}{c^{2} - a^{2}} = \frac{c sin (A - B)}{a^{2} - b^{2}}

a^{2} + b^{2} + c^{2} - a b - b c - c a = \frac{1}{2} [{(b - c)}^{2} + {(c - a)}^{2} + {(a - b)}^{2}]

\frac{a^{2} + b^{2} - c^{2}}{c^{2} + a^{2} - b^{2}} = \frac{tan B}{tan C}

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