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)#