Prove that the points $P( a,\ b+c),\ Q( b,\ c+a)$ and $R( c,\ a+b)$ are Collinear.
Given: Point $P( a,\ b+c),\ Q( b,\ c+a)$ and $R( c,\ a+b)$
To do: To prove that the given points are collinear.
Solution:
Given Points are: $P( a,\ b+c),\ Q( b,\ c+a)$ and $R( c,\ a+b)$
If there are three points $( x_1,\ y_1)\ ( x_2,\ y_2)$ and $( x_3,\ y_3)$. Then the
area of the tringle $=\frac{1}{2}[x_1( y_2-y_3)+x_2( y_3-y_1)+x_3( y_1-y_2)]$
On using the formula,
$= \frac{1}{2}[a( c+a-a-b)+b( a+b-b-c)+c( b+c-c-a)]$
$= ac+a^2-a^2-ab+ab+b^2-b^2-bc+bc+c^2-c^2-ac$
$=0$.
$\because$ Area of the triangle formed by the given points is $0$.
$\therefore$ points $P( a,\ b+c),\ Q( b,\ c+a)$ and $R( c,\ a+b)$ are collinear.
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