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  4. Consider 3 non collinear point A (9,3) ; B (7,-1) and C (1,-1). Let P ( a , b ) be the centre and ' R ' is the radius of the circle 'S' passing through A, B, C. Also H ( barx, bary) are the coordinates of the orthocentre of the triangle ABC whose area be denoted by triangle. If D , E and F are the middle points of BC , CA and AB respectively then the area of the triangle DEF is

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Q. Consider 3 non collinear point $A (9,3) ; B (7,-1)$ and $C (1,-1)$. Let $P ( a , b )$ be the centre and ' $R$ ' is the radius of the circle 'S' passing through A, B, C. Also H ( $\bar{x}, \bar{y})$ are the coordinates of the orthocentre of the triangle $ABC$ whose area be denoted by $\triangle$.
If $D , E$ and $F$ are the middle points of $BC , CA$ and $AB$ respectively then the area of the triangle DEF is

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Solution:

Area of $\Delta DEF =\frac{1}{4} \operatorname{area}(\triangle ABC )=\frac{1}{4}\begin{vmatrix}9 & 3 & 1 \\ 7 & -1 & 1 \\ 1 & -1 & 1\end{vmatrix}=3$