in triangle ABC in which ab=2ac ba isproduced to d and exterior angle cad is bisected by ae - Maths - Quadrilaterals

Question

in triangle ABC in which ab=2ac ba isproduced to d and exterior
angle cad is bisected by ae cutting bc produced in e Prove that c
is the mid point of be

Utsav · Accepted Answer

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Construction- draw CF∥EA intersecting AB at F
To prove C is the midpoint of BE

Now ∠EAD = ∠CFA (1) (corresponding angles as CF∥AB
and AF is a transversal)Also ∠EAC =∠ACE (2) (alternate
angles as CF∥AB and AC is a transversal)From (1) and (2)
∠CFA = ∠ACE (as ∠EAD = ∠EAC = 12∠CAD)Hence in
∆ACF , AC = AF (as in a triangle the sides opposite to equal
angles are equal)Now it is given that AB = 2ACand here AC = AF
Hence AB = 2 AF Thus F is the midpoint of AB Now CF∥AE Hence
∆BCF ~∆BEA(AAA similarity)So BCBE=BFBA⇒BCBE=12(as
F is the midpoint of AB hence BF =12AB)⇒BE=2BCHence C is the
midpoint of BE

in triangle ABC in which ab=2ac . ba isproduced to d and exterior angle cad is bisected by ae cutting bc produced in e. Prove that c is the mid point of be