In the given figure , two triangles ABC and DBC lie on the same base BC . P is a point on BC such PQ || BA and PR || BD . Prove that QR || AD .

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Solution

In triangle ABC $P Q ∥ A B$
Then by BPT theorem,
$\frac{C P}{B P} = \frac{C Q}{A Q}$ ........1
Also in triangle BDC $P R ∥ B D$
Then by BPT theorem,
$\frac{C P}{B P} = \frac{C R}{D R}$ ........2

From eq 1 and eq 2 ,
$\frac{C Q}{A Q} = \frac{C R}{D R}$
Then by reverese of BPT theorem,
$Q R ∥ A D$

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In the given figure , two triangles ABC and DBC lie on the same base BC . P is a point on BC such PQ || BA and PR || BD . Prove that QR || AD .

In triangle ABC PQ∥ABThen by BPT theorem,CPBP=CQAQ ........1Also in triangle BDCPR∥BDThen by BPT theorem,CPBP=CRDR ........2From eq 1 and eq 2 ,CQAQ=CRDRThen by reverese of BPT theorem,QR∥AD