Show that points: A (a, b + c), B (b, c + a), C (c, a + b) are co… - 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 = Modulus of $\frac{1}{2}\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}=\begin{vmatrix}\frac{1}{2}\begin{bmatrix}a&b+c&1\\b&c+a&1\\c&a+b&1\end{bmatrix}\end{vmatrix}$$=\bigg|\frac{1}{2}\left[a(c+a-a-b)-(b+c)(b-c)+1\left\{b(a+b)-c(c+a)\right\}\right]\bigg|$
$=\bigg|\frac{1}{2}\left[a(c-b)-(b^2-c^2)+(ab+b^2-c^2-ac)\right]\bigg|$
$=\bigg|\frac{1}{2}(ac-ab-b^2+c^2+ab+b^2-c^2-ac)\bigg|$
$=\begin{vmatrix}\frac{1}{2}\times0\end{vmatrix}=0$
Therefore, points A, B and C are collinear.

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