In Fig. 6.20, DE || OQ and DF || OR. Show that EF || QR.

We know that,

If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, the other two sides are divided in the same ratio.

In Δ POQ,

DE || OQ [∵ Given]

$\therefore \dfrac{PE}{EQ} = \dfrac{PD}{DO}$ ……….(1)

In Δ POR,

DF || OR [∵ Given]

$\dfrac{PF}{FR} = \dfrac{PD}{DO}$ ……… (2)

From equation (1) and (2), we get :

$\Rightarrow \dfrac{PE}{EQ} = \dfrac{PF}{FR}$

In Δ PQR,

$\Rightarrow \dfrac{PE}{EQ} = \dfrac{PF}{FR}$

We know that,

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

∴ EF || QR.

Hence, proved that EF || QR.