AB and AC are equal chords of a circle. AD and BE are parallel chords through A and B respectively. Prove that AE is parallel to CD.

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- Jul 11th 2010, 05:34 PMLukybearCircle Geometry: Difficult Proving Parrallel Lines
AB and AC are equal chords of a circle. AD and BE are parallel chords through A and B respectively. Prove that AE is parallel to CD.

- Jul 11th 2010, 09:50 PMsimplependulum
Note that if a cyclic quadrilateral is a trapzeium , then it is an isosceles trapzeium . If a pair of the opposite sides are equal in length , then it is also an isosceles trapzeium .

To prove the first line , Let $\displaystyle ABCD $ is a cyclic quadrilateral , assume it is a trapzeium , say $\displaystyle AB \parallel CD $ , then we have $\displaystyle \angle ABD = \angle BDC $ so $\displaystyle \text{arc} AD = \text{arc} BC \implies AD = BC $ thus it is an isos. trapzeium .

I give the proof of the second line to you .

By applying the above property , we find that $\displaystyle DE = AB = AC $ so $\displaystyle AE \parallel CD$ - Jul 12th 2010, 01:00 AMLukybear
Sorry, I am not so sure ABCD is concyclic. As ABCE is the actual quad inside circle.

EDIT: nvm, i have found solution anywaz.