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Math Help - Circle Theorems

  1. #1
    Newbie
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    Mar 2009
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    Circle Theorems

    Hello there,

    I'm having problems visualising this question.
    I have no idea how to prove it, because I don't know what it would look like in the first place... Could somebody please help?

    "The angles of a triangle are 50, 60 and 70 degrees, and a circle touches the sides at A, B, C. Calculate the angles of triangle ABC".

    -------------------------------------------------------------------------------------------------------------------------------

    I have now gotten stuck on another question:

    "Line ATB touches a circle at T and TC is a diameter. AC and BC cut the circle at D and E respectively. Prove that the quadrilateral ADEB is cyclic"

    The problem is that whenever I draw this, points A and B are outside of the circle and thus it cannot be classified as cyclic. Unless I am wrong...
    Last edited by mr fantastic; March 15th 2009 at 03:52 AM. Reason: Merged posts
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, beefsupreme!

    The angles of a triangle are 50, 60 and 70,
    and a circle touches the sides at A, B, C.
    Calculate the angles of triangle ABC.
    Code:
                            P
                            o
                           / \
                          /50\
                         /     \
                        /       \
                       /         \ 
                      /   * * *   \
                     /*           *\
                    *              * 
                 A o- - - - - - - - -o B
                  / \               / \
                 /*  \             /  *\ 
                / *   \     o     /   * \
               /  *    \    O    /    *  \
              /         \       /         \
             /     *     \     /     *     \
            /       *     \   /     *       \
           / 60      *    \ /    *      70 \
        Q o - - - - - - - * o * - - - - - - - o R 
                            C

    We have \Delta PQR with \angle P = 50^o,\,\angle Q = 60^o,\:\angle R = 70^o

    Circle O is inscribed in the triangle
    . . and is tangent to PQ, PR, QR at A,B,C, respectively.

    Draw segments OP, OQ, OR and radii OA, OB, OC.

    OA,OB,OC are penpendicular to the sides of the triangle.

    Since O is the center of the inscribed circle,
    . . OP, OQ, OR bisect \angle P, \angle Q, \angle R, resp.


    We have: . \angle APO = 25^o \quad\Rightarrow\quad \angle POA = 65^o \quad\Rightarrow\quad \angle AOB = 130^o

    Since \Delta AOB is isosceles: . \angle OAB = \angle OBA = 25^o


    Similarly, we find that: . \angle OAC = \angle OCA = 30^o,\:\angle OBC = \angle OCB = 35^o


    Therefore: . \angle A = 55^o,\:\angle B = 60^o,\:\angle C = 65^o

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