D E F are the points on BC-CA-AB respectively of - A B C such that A D - B E - CF are concurrent at O-show that O D - A D - O E - B E - O F - F C - 1

Since DEF are midpoints of the sides #160;Their concurrent point is centroid #160; #160;The centroid divides the median in the ratio 2:1 O ...

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Standard VIII

Mathematics

The Orthocentre

D E F are the...

Question

$D E F$ are the points on $B C, C A, A B$ respectively of $Δ A B C$ such that $A D, B E$ , CF are concurrent at O.show that
$\frac{O D}{A D} + \frac{O E}{B E} + \frac{O F}{F C} = 1$

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Solution

Since $D E F$ are midpoints of the sides
Their concurrent point is centroid
The centroid divides the median in the ratio $2 : 1$
$⟹ \frac{O D}{A D} = \frac{1}{3}$
Similarly, $\frac{O E}{B E} = \frac{O F}{F C} = \frac{1}{3} \frac{O D}{A D} + \frac{O E}{B E} + \frac{O F}{F C} \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1$

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DEF are the points on BC,CA,AB respectively of ΔABC such that AD,BE , CF are concurrent at O.show thatODAD+OEBE+OFFC=1

Since DEF are midpoints of the sides Their concurrent point is centroid The centroid divides the median in the ratio 2:1⟹ODAD=13Similarly,OEBE=OFFC=13ODAD+OEBE+OFFC13+13+13=1

$D E F$ are the points on $B C, C A, A B$ respectively of $Δ A B C$ such that $A D, B E$ , CF are concurrent at O.show that
$\frac{O D}{A D} + \frac{O E}{B E} + \frac{O F}{F C} = 1$