Question

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Answer 1

See diagram.

It is called ladder theorem.

ΔABC and ΔEFC are Similar. (corresponding sides are all parallel)

   => EF/AB = EF/AC    --(1)

ΔACD and ΔAEF are Similar. (corresp. sides are parallel to each other).

   =>  EF/CD = AE/AC    ---(2)

Add the two equations :

    EF [ 1/AB + 1/CD ] =  [ EC + AE] / AC = AC / AC = 1

=>   1/AB + 1/CD = 1/EF

Given AB = a,   CD = b. 

=>   EF = ab/(a+b)