In Fig., D is a point on side BC of △ABC such that BD/CD = AB/AC. Prove that AD is the bisector of ∠BAC.

Dis a point onBCofABC.andBDCD=ABACLet us constructBAtoEsuch thatAE=AC.JoinCE.Now, asAE=AC,BDCD=ABAC⇒BDCD=ABAEAlso,∠AEC=∠ACE(angles opp. to equal sides of a triangle are equal) ... (i)By converse of Basic Proportionality Theorem,DA∥CE∠DAC=∠ACE.. (ii) ...[Alternate angles]∠BAD=∠AEC. (iii) ...[Corresponding ∠s]Also,∠AEC=∠ACE...[From (i)]and∠BAD=∠DAC...[From (ii) and (iii)]So,ADis the bisector∠BAC.