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

Question

Philoid · Accepted Answer

It is visible that ABCE is a cyclic quadrilateral and in a cyclic quadrilateral, the sum of opposite angle is 180⁰

∠AEC = ∠CBA = 180

∠AED = ∠CBA (i)

We know that: Opposite angles of a parallelogram are equal

Hence,

∠ADE = ∠CBA (ii)

Using (i) and (ii)

∠ADE = ∠AED

AE = AD (angles opposite to equal sides are equal)