ABCD is aparallegram. The circle through A,B and C intersects CD produced at E. If AB=10cm, BC= 8cm, CE=14cm. find AE

given: ABCD is a parallelogram. A circle through A, B and C intersects CD produced at E.

AB=10cm, BC= 8cm, CE=14cm. 

to find AE

∠AED+∠ABC=180 deg ..........(1)  [sum of opposite angles of the concyclic quadrilateral is supplementary]

∠EDA+∠ADC=180 deg ..........(2)  [angles formed on the same side of  a straight line]

∠EDA+∠ABC=180 deg............(3) [since ∠ADC=∠ABC opposite angles of a parallelogram are equal]

from (1) and (3)

∠AED+∠ABC=∠EDA+∠ABC

∠AED=∠EDA  ................(4)

now in the triangle AED,

from (4) AE=AD [sides opposite to equal angles are equal ]

hence AE=AD=BC=8 cm [AD=BC opposite sides of a parallelogram are equal]

hope this helps you.