Question

Point D and E lie on side AB of ∆ABC such that AD = BE. If DP || BC and EQ || AC then prove that PQ || AB.

Answer 1

Given : ∆ABC in which

AD = BE,

DP || BC and EQ || AC

To prove PQ || AB

Proof : In ∆ABC DP || BC

AD/BD = AP/CP …..(i) (By B.P. Theorem)

and QE || AC

BE/AE = BQ/QC …..(ii)

∵ AD = BE (given) ……(iii)

AD + DE = BE + DE

AE = BD …..(iv)

putting values from equation (iii) and (iv) in (ii)

BE/AE = AD/BD = BQ/QC

From equation (i) and (v)

AP/DB = AP/CP = BQ/QC

⇒ AP/CP = BQ/QC

⇒ CP/AP = QC/BQ

⇒ PQ || AB