Show that points A( a, b + c), B(b, c + a), c(c, a + b) are colli… - Vidyadip

Question

Show that points A( a, b + c), B(b, c + a), c(c, a + b) are collinear.

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Answer

Area of triangle ABC = $\frac{1}{2}\left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1 \\ {{x_2}}&{{y_2}}&1 \\ {{x_3}}&{{y_3}}&1 \end{array}} \right| = {\frac{1}{2}\left[ {\begin{array}{*{20}{c}} a&{b + c}&1 \\ b&{c + a}&1 \\ c&{a + b}&1 \end{array}} \right]} $

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

$= {\frac{1}{2}\left[ {a\left( {c - b} \right) - \left( {{b^2} - {c^2}} \right) + \left( {ab + {b^2} - {c^2}- ac} \right)} \right]} $

$ = {\frac{1}{2}\left( {ac - ab - {b^2} + {c^2} + ab + {b^2} - {c^2} - ac} \right)} $

$= {\frac{1}{2} \times 0} = 0$

Therefore, points A, B and C are collinear.

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