Question

abc is a triangle in which ab=ac if the bisectors of angle b and angle c meet ac and ab in d and e respectively prove that bd = ce​

Answer 1

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Question



In a  ΔABC,AB=AC.  If the bisectors of  ∠B  and ∠C  meet  AC  and  AB  at points  D  and  E  respectively, show that : 

(i) ΔDBC≅ΔECB

(ii) BD=CE





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Solution



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AB=AC

EC is bisector of ∠C

BD is bisector of ∠B

(i)  AB=AC→∠B=∠C  (isosceler triangle)

in ΔDBC and ΔECB

BC→commonside ..(1)

∠DBC=1/2∠B=1/2∠C     (∵∠B=∠C)

∠ECB=1/2∠B

⇒∠DBC=∠ECB ...(2)

∠EBC=∠B=∠C=∠DBC

⇒∠EBC=∠DCB ...(3)

From (1), (2) & (3)

ΔDBC≅ΔECB 

⇒BD=CE     (∵ΔDBC≅ECB)