In the given figure, \(P\), \(Q\) and \(R\) are points on \(OA\), \(OB\) and \(OC\) respectively such that \(PQ||AB\) and \(PR||AC\). Show that \(QR||BC\).
Proof:
In \(\Delta OAB\),
\(PQ || AB\)
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
\(\frac{OP}{PA} =\) - - - - (1)
In \(\Delta OAC\),
\(PR || AC\)
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
\(\frac{OR}{RC} = \) - - - - (2)
From (1) and (2),
\(\frac{OR}{RC} = \)
In \(\Delta OBC\),
\(\frac{OR}{RC} = \)
If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Thus, \(QR||BC\).
Hence proved.