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Math Help - geo circle problem

  1. #1
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    geo circle problem

    please help!!!

    In the diagram attached, the points A, B, D and E lie on a circle.
    AE = BE = BC.
    The lines BE and AD intersect at F.
    Angle DCB = x.

    Find the size of angle AEB in terms of x.
    You must give a geometric reason for each step leading to your answer.
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  2. #2
    MHF Contributor
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    I can see no diagram in the attached file, or my browser cannot show what is on the attached file.

    If you can fix that, so that we can see the figure, then your Problem might be solvable. I have no time now. I took a peek only. I'd continue later on, some 7-8 hours from now.
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  3. #3
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    Here is the diagram for those who cannot see it.
    Attached Thumbnails Attached Thumbnails geo circle problem-dia.gif  
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  4. #4
    MHF Contributor
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    So that is the figure. [Thanks, Plato.]

    Okay.

    So AE = BE = BC.
    That means we have two isosceles triangles here.

    ---------------------------------
    In isosceles triangle BCE,

    angle BEC = angle BCE = x degrees
    Reason: The base angles of an isosceles triangle are equal in measure. (If there is no geometric rule/theorem/conhrcture/whatever for that, we can prove that geometrically.)

    -----------------------------------
    So, in isosceles triangle BEA,

    angle EBA = 2x degrees
    Reason: In a triangle, an exterior angle is equal in measure to the combined measurements of the other two interior angles.
    No such rule/whatever again?
    Then angle EBC = 180 -2x -----there are 180 degrees in a triangle.
    So angle EBA = 180 -(180 -2x) = 2x dgrees. ----a straight angle is 180 degrees in measure.

    Hence, angle EAB = angle EBA = 2x degrees
    Reason: Again, the base angles of an isosceles triangle are equal in measurements.

    Therefore, angle AEB = 180 -2x -2x = (180 -4x) degrees. -----answer.
    Why?
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