ABCD is a parallelogram. The circle through A, B, and C intersects CD (produced if necessary) at E. Prove that AE=AD.

From the figure,⇒∠ADE+∠ADC=180o(Linear pair) ... (i)Also,□ABCEis a cyclic quadrilateral.So,∠AED+∠ABC=180o... (ii) (opposite angles of cyclic quadrilateral sum upto180o)∠ADC=∠ABC... (iii) (Opposite angles of a parallellogram are equal)Using (i), (ii) and (iii),∠AED+∠ABC=∠ADE+∠ABC⇒∠AED=∠ADENow, In△AED,∠AED=∠ADE⇒AD=DE(Sides opposite to equal angles)