The straight lines EF and ED meet at E. B and C are fixed points on EF and A is a variable point on ED. Determine the position of A, in terms of ∠DEF, EB and EC, such that AB + AC is a minimum? [You are required to find EA at this minimum].

1 Answer
Oct 15, 2017

see explanation.

Explanation:

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See Fig2.
Reflect point C over line ED to get C',
so long as A moves along line ED, => AC=AC',
=> DeltaEC'A and DeltaECA are congruent,
when B,A,C' lie on a straight line, minimum value of AB+AC (=AB+AC') can be obtained.
let angleDEF=x, => angleDEC'=x,
and let |ABC| denote area of DeltaABC,
Now, |EC'A|=|ECA|,
=> |EC'A|+|EAB|=|ECA|+|EAB|,
=> |EC'B|=|ECA|+|EAB|
=> 1/2*EB*EC'*sin2x=1/2*EC*EA*sinx+1/2*EB*EA*sinx,
=> EB*EC*(sin2x)/sinx=EC*EA+EB*EA,
=> EB*EC*(2sinxcosx)/sinx=EA*(EC+EB),

=> EA=(2*EB*EC*cosx)/(EB+EC)

=> EA=(2*EB*EC*cosangleDEF)/(EB+EC)