Question

In the given figure, circles with centres C and D touch internally at point E. D lies on the inner circle. Chord EB of the outer circle intersects inner circle at point A. Prove that, seg EA ≅ seg AB.

Answer 1

Circles with centres C and D touch internally at point E.
Join ED.
By theorem of touching circles, points E, C and D are collinear. 

Since D lies on the inner circle with centre C, therefore, ED is the diameter of the inner circle.
∴ ∠EAD = 90º       (Angle inscribed in a semi-circle is a right angle)
EB is the chord of the outer circle with centre D.
∴ Point A is the mid-point of chord EB.            (Perpendicular drawn from the centre of a circle on its chord bisects the chord)
⇒ seg EA ≅ seg AB