in a triangle ABC, AB=AC.AB is produced to D such that BD=BC.show that angle ACD = (Angle ADC *3)



In ADC, DE bisects D, soADDC = AEEC   angle bisector theorem   ...1In ABC, BE bisects B, soABBC = AEEC   angle bisector theorem   ...2from 1 and 2,ADDC =ABBC 

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In triangle BDC, BD = BC
so <BDC = <BCD

or <ADC = <BCD -------(1)

In triangle ABC, AB = AC
so <ABC = <ACB   ------(2)

also,   < A + <ABC + < ACB = 180  -----(3)

In triangle ADC,
<A + <ADC + <ACD = 180   -------(4)

from (3) and (4)
< A + <ABC + < ACB = <A + <ADC + <ACD
or <ABC + < ACB = <ADC + <ACD
or 2 <ACB = <ADC + <ACD    (from (2)
or 2 (<ACD - <BCD) = <ADC + <ACD
or 2 (<ACD - <ADC) = <ADC + <ACD   (from (1)
or 2 <ACD - 2<ADC = <ADC + <ACD
or 2 <ACD - <ACD = <ADC + 2<ADC
or <ACD = 3 <ADC  [proved]
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