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Math Help - Show that line passes through a point

  1. #1
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    Show that line passes through a point

    Given is a triangle ABC. A circle with center D is tangent to BC, AB and AC at points that lie outside the given triangle. Show that AD passes through the circumcircle of triangle BCD.

    I have no idea how to attempt this, and to begin with I don't really know the method of construction of such a circle, ie. how do I make a circle that is tangent to three given lines?

    Thanks for help.
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  2. #2
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    Quote Originally Posted by atreyyu View Post
    Given is a triangle ABC. A circle with center D is tangent to BC, AB and AC at points that lie outside the given triangle. Show that AD passes through the circumcircle of triangle BCD.

    I have no idea how to attempt this, and to begin with I don't really know the method of construction of such a circle, ie. how do I make a circle that is tangent to three given lines?

    Thanks for help.
    Hi atreyyu,

    if you extend the three sides of the triangle such that the triangle sides
    are now line segments of these lines, then, there are 3 circles that touch one side of the triangle tangentially
    and touch the extended lines tangentially.
    If you bisect the appropriate external angle you will have a line whose points are
    the same perpendicular distance from the extended line and a side of the triangle.
    Do the same for the other relevant extended line and you will find the centre of the circle with centre D (of which there are 3).

    That can get you started geometrically.
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  3. #3
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    Could you show this step by step? I don't really see it.
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  4. #4
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    Let AB touches the circle at E, BC at F and AC at G. Since AE and AG are the tangents to the circle, they are equal.
    AD is the angle bisector of A and DE is perpendicular to BC, hence AD is perpendicular bisector of BC. So the circumcenter of the circle passing through BCD lies on AD.
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  5. #5
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    Hello, atreyyu!

    A sketch would help . . .


    Given is a triangle ABC.
    A circle with center D is tangent to AB, BC, AC
    at points that lie outside the given triangle.

    Show that AD passes through the circumcircle of \Delta BCD.

    This is one possible diagram . . .


    Code:
                         \
                  * * *   \
              *           *\
            *               *
           *                 *
                              \   /
          *                   *\ /
          *         *         * o B
          *         D         */ \
                              /   \
           *                 *     \
            *               *       \
              *           */         \
            - - - * * * - o - - - - - o - - -
                         /C           A\
                        /               \

    Circle D is tangent to side BC
    . . and tangent to sides AB and AC extended.

    (There are two more locations for circle D.)



    They could have said: "Circle D is externally tangent to triangle ABC."
    . .
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  6. #6
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    And could you explain step-by-step how to construct that circle with compass/straightedge?
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  7. #7
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    The attachment is a simplistic version,
    for a right-angled isosceles triangle.
    Unfortunately, the labelling does not match sa-ri-ga-ma's post.

    You may bisect 2 external angles,
    or one of them and the angle at A.
    They intersect at the same point.

    Draw the 2 green circles at the vertices B and C.
    Bisect those external angles with the purple circles and pair of light-blue circles
    (you may prefer arcs).
    The purple circles are the same size.
    The light-blue circles are also of equal size.

    The centres are marked.
    Draw the bisectors through the points of intersection of the purple circles
    and the points of intersection of the light-blue circles.

    The bisectors intersect at D.

    It's the same procedure for other triangles.
    Attached Thumbnails Attached Thumbnails Show that line passes through a point-bisector-external-angles.jpg  
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  8. #8
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    Thanks a lot, I finally drew the darn thing.
    Could you give me a hand with the proof? The thing seems obvious when I look at it now, but I still don't know how to proceed.
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  9. #9
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    If we bisect angle BAC, the centre D of the large externally tangential circle
    lies on this line also.

    We find the centre of a circumcircle by bisecting the sides of the triangle housing it.

    This means the centre of the circumcircle of triangle BDC is on the bisector of side BC.

    The centre D of the external circle also lies on this line.

    This is a very simple and convenient case, however
    and there is more to the general case.

    Can you draw a scalene triangle and work out a proof for that,
    using these posts as a guide?
    Attached Thumbnails Attached Thumbnails Show that line passes through a point-bisectors-circumcircle.jpg  
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  10. #10
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    Eh, nope. When the triangle gets scalene, my reasoning is laid waste.
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  11. #11
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    It involves drawing a triangle with 3 sides of different lengths.
    Extend the sides of the triangle out again.
    Bisect angles, find the centre of the externally tangential circle.
    Bisect the side of the scalene triangle that the external circle touches.
    The centre of the circumcircle lies on that bisector.

    Almost exactly as already shown.
    Please try.
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  12. #12
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    OK, OK. I have already managed to draw a scalene triangle, etc etc. I see it that the centre of the circumcircle lies on the bisector. But what you wrote before is in no way relevant to a scalene triangle, as, forn instance, bisector of BC is not the bisector of angle at A anymore.
    I really don't know what do now.
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  13. #13
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    Quote Originally Posted by atreyyu View Post
    OK, OK. I have already managed to draw a scalene triangle, etc etc.

    there is no need to be impatient

    I see it that the centre of the circumcircle lies on the bisector. But what you wrote before is in no way relevant ?? to a scalene triangle, as, for instance, bisector of BC is not the bisector of angle at A anymore.

    Yes, that's true, I thought it might be helpful to leave you something to do,
    instead of just giving you the answer.
    From which you would have learnt nothing.


    I really don't know what do now.
    Can you see that the line AD must of necessity cross through the circumcircle of BCD,
    since it is between the lines BD and CD ?

    Note that the side BC is a chord of the circumcircle.

    That's all you were asked to prove.
    I don't know if you'd be interested in examining the other 2 external circles ??
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