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Math Help - one angle one circle and one point

  1. #1
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    Exclamation one angle one circle and one point

    hi to everyone.

    here I have a problem which has to be solved geometrically. The problem is the following:

    There exists an angle. we have a point "A" which is set in a random place inside the angle. The problem is to draw a circle inside the angle which passes through the point "A" and is tangent over two lines of the angle. In other words, the circle passes through three points. two points are over the two lines of the angle and one point is "A"
    How to find the center of circle and its radius???

    I have also attached one picture so you can imagine the situation better.

    Thanks in advance for your help.

    Attached Thumbnails Attached Thumbnails one angle one circle and one point-picture.jpg  
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  2. #2
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    Hello, Narek!

    I don't have it solved yet, but I have an observation . . .


    Given a point A interior to angle POQ.
    Find the center and radius of the circle which passes through A
    and is tangent to the sides of the angle.
    Code:
                          P
                         /
                        /   * * * 
                       /*           *
                    E *               *
                     *                 *
                    /   *
                   /*      *  C         *
                  / *         o         *
                 /  *         * *       *
                /             *   *
               /     *        *     *  *
              /       *       *       o A
             /          *     *     *
          O * - - - - - - - * * * - - - - Q
                              D

    The center lies on the angle bisector OC\!:\;\;CD = CE.

    The center is also equidistant from point A (focus) and a line OQ (directrix).
    . . This determines a parabola.

    The center is the intersection of this parabola and the angle bisector,
    . . and there are two such points.


    Maybe someone can use this knowledge . . . or not.

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  3. #3
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    any other help people ?
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  4. #4
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    Hi

    The first step is to draw a circle tangent to the angle
    Take any point M1 on the first line and take the point N1 on the second line such as ON1=OM1
    Then draw the 2 perpendicular lines at M1 and N1. They cross at O1 center of the circle C1
    Draw the bisector of the angle
    C1 is tangent to the angle at M1 and N1
    Here is the situation (if you are very lucky then A is on C1 and you are almost done ; almost because there are 2 circles that answer the problem as already highlighted by Soroban)


    Now draw the line (OA)
    It cuts C1 in 2 points B1 and D1
    Let h1 be the homothety (center O) that transforms B1 into A
    It transforms C1 into a circle C2 (center O2)
    Homothety transforming any line into a parallel line, C2 is tangent to the angle and passes through A
    (O1B1) is transformed into (O2A) which is parallel to (O1B1)
    Draw the line parallel to (O1B1) that passes through A. It cuts the bissector at O2
    Then you get the first circle (see picture)
    Make the same with the homothety (center O) that transforms D1 into A and you will get the second circle



    Note that this method is OK for any A except if A is on the bisector. In this case you can manage by building the parallel to (B1M1) that passes through A. It cuts the first line of the angle in M2 which is the tangent point of the circle you are looking for. And so on ...
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  5. #5
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    Nice solution, r-g!
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  6. #6
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    Quote Originally Posted by Opalg View Post
    Nice solution, r-g!
    Thanks a lot
    I could not find the solution yesterday evening
    I think that my brain ran all night long because this morning I found it !
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  7. #7
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    Quote Originally Posted by Narek View Post
    any other help people ?
    It would have been polite to have added a thankyou to Soroban for his input ....
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  8. #8
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    Wink

    Quote Originally Posted by mr fantastic View Post
    It would have been polite to have added a thankyou to Soroban for his input ....
    Don't be sad my friend. I add all the "Thank You"s at last
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  9. #9
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    Thumbs up Thank you all

    Thank you people for your help. This was kind of you
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  10. #10
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    What mr. fantastic meant was clicking on the "thanks" button at the bottom of Soroban's post. That adds to Sorobans "number of times thanked" total which just posting "thank you" does not.
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  11. #11
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    Quote Originally Posted by HallsofIvy View Post
    What mr. fantastic meant was clicking on the "thanks" button at the bottom of Soroban's post. That adds to Sorobans "number of times thanked" total which just posting "thank you" does not.
    dear friend,

    can't you see the "Thank You" in the bottom of the post? don't you think you must review the posts before you add your personal opinion?? and I completely understand the differences between words and button and their functionalities. I am also not bad in understanding English language, so I don't think I need someone to explain for me what is said to me
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  12. #12
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    Quote Originally Posted by HallsofIvy View Post
    What mr. fantastic meant was clicking on the "thanks" button at the bottom of Soroban's post. That adds to Sorobans "number of times thanked" total which just posting "thank you" does not.
    Actually what I was commenting on was that the counter-reply

    Quote Originally Posted by Narek
    any other help people ?
    straight after Soroban's post should have contained some acknowledgement to Soroban ....
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