Question

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].

Answer 1

see explanation.

Explanation:

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)`