3rd selection: ΔEAC is equilateral triangle.
You need to show AB ≅ ED. Knowing AEC is isosceles with base EC (2nd selection) is equivalent to knowing ∠E ≅ ∠C (1st selection). All that tells you is AC ≅ AE, which is not what you need. Likewise, the 4th selection tells you AE ≅ CE, which doesn't help, either (for the same reason).