# ΔABC and ΔDBC lie on the same side of BC, as shown in the figure. From a point, P on BC, PQ || AB and PR || BD are drawn, meeting AC at Q and CD at R respectively. Prove that QR || AD.

We can observe two triangles in the figure.

In ∆ABC,

PQ AB

Applying Thale’s theorem, we get

…(i)

In ∆BDC,

PR BP

Applying Thale’s theorem, we get

…(ii)

Comparing equations (i) and (ii),

Now, applying converse of Thale’s theorem, we get

Hence, QR is parallel to the AD.

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