Math CentralQuandaries & Queries


Question from Tim, a student:

Hi! I've been working on this for a while and I'm quite stuck. If anyone can help that would be great.

The sides BC and AD of a quadrilateral ABCD are parallel. A circle meets the side AB at B and E and the side CD at C and F. Prove that the quadrilateral AEFD is cyclic.


You should know the theorem:

If the points PQRS lie in that order on a circle then each pair of opposite angles add up to 180 degrees, and conversely, if in a quadrilateral PQRS the angles at P and at R sum to 180 then the four points lie in the order P,Q,R,S around a circle.

The statement of your problem says that BCFE lie on a circle in that order, so the angles at B and F sum to 180; it is your job to prove that in quadrilateral AEFD the angles at F and at A sum to 180.


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