In triangle ABC, AB = AC. The point D lies on AB and the point E lies on AC with AD = AE, and the segments BE and CD cross at O. Which one of these does NOT follow?

Triangles ABE and ACD have AB = AC, AE = AD and the angle at A in common, so they are congruent by SAS; CPCT gives BE = CD and ∠ABE = ∠ACD. Since ∠ABC = ∠ACB, subtracting equals leaves ∠OBC = ∠OCB, so triangle OBC is isosceles and OB = OC. Also AD : AB = AE : AC, so DE ∥ BC by the converse of the basic proportionality theorem. But OD = OB is not forced: take AB = AC = √13, BC = 4 and AD = AE = 1, and the crossing point gives OB ≈ 2.6 while OD ≈ 0.7. The equal segments BE and CD are cut by O into pieces that need not match up.