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Topic: Question about trapezoids
Replies: 7   Last Post: Jul 8, 2008 5:13 AM

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Peter Chang

Posts: 8
From: new zealand
Registered: 7/5/08
Re: Question about trapezoids
Posted: Jul 8, 2008 5:12 AM
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In fact the theorem we are to approve is totally wrong. We can disprove it easily.

In an isosceles ABC, Dram line segment BE first. When drawing line CD , you will find you can draw two lines that are equal to BE.

Draw a circle with radius CD, and C as its center. As generally CD is not perpendicular to AB, the circle will intersect line AB twice, at D and another point, say D'. CD = CD'.

Now connect E and D, E and D'. If ED' is parallel to BC, then quadrilateral BCED' is cyclic, and BCED is not. If ED is parallel to BC, then quadrilateral BCED is cyclic, and BCED' is not.

So now consider the following questions:
(1) In isosceles ABC, DC = BE, is quadrilateral ABCD cyclic?
(2) In isosceles ABC, D'C = BE, is quadrilateral ABCD' cyclic?

Note that to make the theorem true, we need to add another condition: CD and EB are perpendicular to AB and AC respectively.

Message was edited by: Peter Chang

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