# Circle Geometry: Difficult Proving Parrallel Lines

• Jul 11th 2010, 05:34 PM
Lukybear
Circle 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 PM
simplependulum
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 AM
Lukybear
Sorry, I am not so sure ABCD is concyclic. As ABCE is the actual quad inside circle.

EDIT: nvm, i have found solution anywaz.