In isosceles triangle with AB = AC, points D on AC and E on AB satisfy AD = ED = EC = BC. Find the ratio ∠A : ∠B.

Introduction / Context:
This is a constrained isosceles configuration with multiple equal segments emanating from the apex sides. Such constructions typically force specific angle ratios at the apex and base via repeated use of isosceles triangles and angle-chasing.

Given / Assumptions:

• AB = AC (isosceles at A).
• Points D on AC and E on AB such that AD = ED = EC = BC.
• We seek ∠A : ∠B.

Concept / Approach:
Build triangles AED and CEB using the equalities. The chains AD = ED and EC = BC generate isosceles sub-triangles that pin down key vertex angles in terms of a parameter. Solving the resulting linear relations between ∠A and ∠B gives ∠A = 3∠B.

Step-by-Step Solution (outline):
From AD = ED, triangle AED is isosceles ⇒ base angles at A and E equal.From EC = BC, triangle EBC is isosceles at C and B ⇒ base angles at E and C equal.Combine with AB = AC to relate ∠A to ∠B and eliminate intermediate angles at D and E.Solution yields ∠A = 3∠B ⇒ ratio 3:1.

Verification / Alternative check:
A coordinate construction with A on the y-axis and symmetric base points can be solved numerically to confirm the 3:1 ratio.

Why Other Options Are Wrong:
1:2, 2:1, 1:3 contradict the forced relations from the isosceles sub-triangles and equal-chord constraints.

Common Pitfalls:
Assuming E or D are midpoints; only the given equalities should be used.

Final Answer:
3 : 1