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Math Help - cyclic quadrilateral ...

  1. #1
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    cyclic quadrilateral ...

    if angles DAC , BAC
    are equal and DC = CB
    show that ABCD is a
    cyclic quadrilateral.

    cyclic quadrilateral ...-cyclic-quadrilateral.jpg
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  2. #2
    Member rtblue's Avatar
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    Are you familiar with the angle bisector theorm? Well, by this theorm, once we recognize AC as the angle bisector, we can set up a proportion: \displaystyle \frac{AD}{DC}=\frac{AB}{BC}.

    Since we have DC=BC (let's call these two sides x) we have \displaystyle \frac{AD}{x}=\frac{AB}{x} \Rightarrow AB=AD

    Now, by definition, we have ABCD as a kite. Look up some properties of a kite, and see if you can go on to prove that it is a cyclic quadrilateral. (Hint: It will be cyclic if it is formed by two right triangles)
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  3. #3
    Senior Member BAdhi's Avatar
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    rtblue, can't we come to your conclusion (i.e. AB=AD) in the following way as well?,

    since the two triangles are having an equal angle and two equal sides, all the charasteristics of the two triangles are equal
    (I don't know the english word for these triangles, if some one know please let me know).

    if this is true,

    i think we cannot prove that this is a cyclic quadrilateral. Because if this is a cyclic quadrilateral, due to symmetricity, AC has to be the diammeter of the circle. Which makes \angle ADC=\angle ABC=90^\circ

    I cannot find enough information to show this
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  4. #4
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    Quote Originally Posted by rtblue View Post
    Are you familiar with the angle bisector theorm? Well, by this theorm, once we recognize AC as the angle bisector, we can set up a proportion: \displaystyle \frac{AD}{DC}=\frac{AB}{BC}.
    Shouldn't DCB be a straight line to apply the angle bisector theorem? And AB does not have to be equal to AD: look at this picture.



    Quote Originally Posted by BAdhi
    since the two triangles are having an equal angle and two equal sides, all the charasteristics of the two triangles are equal
    (I don't know the english word for these triangles, if some one know please let me know).
    The angle x is not located between the two equal sides. For the terminology see the sufficient conditions for triangle congruence.
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  5. #5
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    Quote Originally Posted by rtblue View Post
    Are you familiar with the angle bisector theorm? Well, by this theorm, once we recognize AC as the angle bisector, we can set up a proportion: \displaystyle \frac{AD}{DC}=\frac{AB}{BC}.

    Since we have DC=BC (let's call these two sides x) we have \displaystyle \frac{AD}{x}=\frac{AB}{x} \Rightarrow AB=AD

    Now, by definition, we have ABCD as a kite. Look up some properties of a kite, and see if you can go on to prove that it is a cyclic quadrilateral. (Hint: It will be cyclic if it is formed by two right triangles)
    This theorem does not work in this situation

    Can you prove that AD=AB ?
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  6. #6
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    The yellow triangles are congruent in the attached construction.
    If you label the angles in these,
    you can prove that the opposite angles of the quadrilateral sum to 180^o
    Attached Thumbnails Attached Thumbnails cyclic quadrilateral ...-cyclic-quad.jpg  
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  7. #7
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    I see a kite. A kite is cyclic if and only if it hastwo right angles


    bjh
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  8. #8
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    The circle for which the concyclic points A, B, C and D are on the circumference
    does not need to have it's centre on [AC].

    Neither does the quadrilateral have to have an axis of symmetry.
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  9. #9
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    cyclic quadrilateral

    Quote Originally Posted by Archie Meade View Post
    The circle for which the concyclic points A, B, C and D are on the circumference
    does not need to have it's centre on [AC].

    Neither does the quadrilateral have to have an axis of symmetry.

    A kite is a special quadrilateral and always has an axis of symmetry It is cyclic when formed by two congruent right triangles



    bjh
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  10. #10
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    Yes, but razemsoft's image is not fully defined in such a way.
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  11. #11
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    Hi Archie Meade,
    From the OP's givens and drawing proof that this is a kite is relatively easy.If you would like one I will reply


    bjh
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  12. #12
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    Why don't you ask razemsoft instead of assuming ?
    There is the possibilty that other possible cases are not eliminated.

    In the sketch I uploaded, the yellow triangles could have an extremely small vertical height
    with the circles practically overlapping.
    I've exaggerated it for illustration.
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  13. #13
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    Quote Originally Posted by Archie Meade View Post
    Why don't you ask razemsoft instead of assuming ?
    There is the possibilty that other possible cases are not eliminated.

    In the sketch I uploaded, the yellow triangles could have an extremely small vertical height
    with the circles practically overlapping.
    I've exaggerated it for illustration.
    The question doesn't assume that the shape is a kite.
    it assumes 4 pts. ABCD which have the following conditions:

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  14. #14
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    As the outer circle's radius is reduced
    and it overlaps the inner one, the angle \omega\rightarrow\ 0

    Then we have a quadrilateral with two 90^o edges,
    but in the general case of \omega>0, the quadrilateral is also cyclic.
    Attached Thumbnails Attached Thumbnails cyclic quadrilateral ...-mhf-cyclic-quad2.jpg  
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  15. #15
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    Can you prove that:

    AC is perpendicular to BD

    OR

    AC divides BD into 2 equal parts

    OR

    The angels ACB and ACD are equal ?
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