Results 1 to 4 of 4

Math Help - Cyclic quadrilateral

  1. #1
    Newbie
    Joined
    Feb 2009
    Posts
    3

    Cyclic quadrilateral

    1. ABCP are points on the circle.ABC is a isocelels triangle with AB=AC and angle bisector of B meets circle at P. If AP and BC is extended to meet at Q, show that AC=QC.
    Please let me know the steps to solove the above problem
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,252
    Thanks
    5
    \widehat{ABC}=\widehat{ACB}

    \widehat{ABP}=\widehat{PBC}=\widehat{PAC}=\frac{1}  {2}\widehat{ACB}

    \widehat{ACB}=\widehat{PAC}+\widehat{AQC}=\frac{1}  {2}\widehat{ACB}+\widehat{AQC}\Rightarrow

    \Rightarrow\widehat{AQC}=\frac{1}{2}\widehat{ACB}=  \widehat{PAC}\Rightarrow\Delta ACQ isosceles \Rightarrow AC=CQ
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,707
    Thanks
    625
    Hello, reshma!

    1. ABCP are points on the circle.
    ABC is an isocelels triangle with AB=AC, and angle bisector of B meets circle at P.
    If AP and BC is extended to meet at Q, show that: AC=QC.

    Since \angle ABC = \angle ACB,\;\text{arc}(AC) = \text{arc}(AB)

    Since \angle ABP = \angle PBC,\;\text{arc}(AP) = \text{arc}(PC)

    Let \text{arc}(AP) \,=\,\text{arc}(PC) \,=\,a
    . . Then: . \text{arc}(AB) \,=\,2a


    \angle PAC is an inscribed angle.
    . . It is measured by: . \tfrac{1}{2}\text{arc}(PC) \,=\,\tfrac{1}{2}a

    \angle AQB is formed by two secants.
    . . It is measured by: \tfrac{1}{2}\left[\text{arc}(AB) - \text{arc}(PC)\right] \;=\;\tfrac{1}{2}\left[2a-a\right] \;=\;\tfrac{1}{2}a

    Hence: . \angle PAC \,=\,\angle AQB, and \Delta ACQ is isosceles.

    Therefore: .  AC \,=\,CQ

    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Feb 2009
    Posts
    3

    doubt

    Hi,

    Can u pls explain me the below part:

    is formed by two secants
    It is measured by:

    Thanks in advance
    Reshma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cyclic quadrilateral.
    Posted in the Geometry Forum
    Replies: 3
    Last Post: July 29th 2011, 04:01 AM
  2. Is a quadrilateral cyclic?
    Posted in the Geometry Forum
    Replies: 2
    Last Post: March 21st 2010, 01:01 AM
  3. Cyclic Quadrilateral
    Posted in the Geometry Forum
    Replies: 2
    Last Post: March 17th 2010, 01:09 AM
  4. cyclic quadrilateral
    Posted in the Geometry Forum
    Replies: 2
    Last Post: November 21st 2009, 09:09 AM
  5. cyclic quadrilateral
    Posted in the Geometry Forum
    Replies: 3
    Last Post: September 14th 2009, 05:41 AM

Search Tags


/mathhelpforum @mathhelpforum