A rectangle ABCD is inscribed in a circle with centre O. Its diagonal CA is produced to a point E, outside the circle. ED is a tangent to the circle at D. If AC=2BC, then what is the measure of ∠DEC ?

In a rectangle ABCD inscribed in a circle, diagonal AC passes through centre O (property of cyclic rectangles). Given AC = 2BC, by Pythagoras: AC² = AB² + BC² → 4BC² = AB² + BC² → AB = BC√3. This gives ∠BCA = 30° (since tan ∠BCA = BC/AB = 1/√3). Since CA is produced to E, points D-C-E are collinear. Angle between tangent ED and chord ED equals angle in alternate segment: ∠EDC = ∠DCA = 30°. In triangle DCE: ∠DCE = 180° - 60° = 120° (external angle), so ∠DEC = 180° - 120° - 30° = 30°. The answer 30° is option A.