Question

Show that the points Ala, b + c), Blb, c + a) and Clc, a + b) are collinear.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given points are (a, b + c), (b, c + a) and (c, a + b).

Let consider,

$\rm :\longmapsto\: \begin{gathered}\sf \left | \begin{array}{ccc}a&b + c&1\\b&c + a& 1\\c& a + b&1\end{array}\right | \end{gathered}$

$\boxed{ \tt{ \: OP \:C_2 \: \to \: C_2 \: + \: C_1 \: }}$

$\rm \:  =  \: \: \begin{gathered}\sf \left | \begin{array}{ccc}a&b + c + a&1\\b&c + a + b& 1\\c& a + b + c&1\end{array}\right | \end{gathered}$

$\red{\rm :\longmapsto\:Taking \: a + b + c \: common \: from \: C_2 \: }$

$\rm \:  =  \: \: (a + b + c)\begin{gathered}\sf \left | \begin{array}{ccc}a&1&1\\b&1& 1\\c& 1&1\end{array}\right | \end{gathered}$

We know,

If two rows or columns are identical, the determinant value is zero.

So,

$\rm \:  =  \: (a + b + c) \times 0$

$\rm \:  =  \: 0$

So,

$\rm :\longmapsto\: \begin{gathered}\sf \left | \begin{array}{ccc}a&b + c&1\\b&c + a& 1\\c& a + b&1\end{array}\right | \end{gathered} = 0$

Or

$\rm :\longmapsto\: \dfrac{1}{2} \begin{gathered}\sf \left | \begin{array}{ccc}a&b + c&1\\b&c + a& 1\\c& a + b&1\end{array}\right | \end{gathered} = 0$

This implies, area of triangle formed with the vertices (a, b + c), (b, c + a) and (c, a + b) is 0.

Hence,

$\boxed{ \tt{ \: (a, b + c), (b, c + a) \: and \: (c, a + b) \: are \: collinear \: }}$

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More to Know :-

1. The determinant value remains unaltered if rows and columns are interchanged.

2. The determinant value is 0, if two rows or columns are identical.

3. The determinant value is multiplied by - 1, if successive rows or columns are interchanged.

4. The determinant value remains unaltered if rows or columns are added or subtracted.