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Math Help - Geometry Construction... circles

  1. #1
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    Geometry Construction... circles

    Given an angle BAC, and a point D inside it, construct a circle passing through D and tangent to the sides of the angle.
    Geometry Construction... circles-bac.jpg
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  2. #2
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    Quote Originally Posted by MATNTRNG View Post
    Given an angle BAC, and a point D inside it, construct a circle passing through D and tangent to the sides of the angle.
    Click image for larger version. 

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    Hi MATNTRNG,

    First, construct the bisector of the angle.
    This is the blue line in the attachment.

    Then draw a line from the point of intersection of the 2 given lines to the given point (the red one).

    Next, draw any circle with centre on the bisector (the purple one).
    Then draw the green line from the centre of this circle to the point where the circle intersects the red line.

    A parallel line from the given point to the angle bisector locates the centre of
    one of the required circles.

    It is simply an enlargement.

    To get the other one, join the centre of the purple circle to the 2nd point
    of contact with the red line and draw a parallel line from
    the given point to the angle bisector.
    Attached Thumbnails Attached Thumbnails Geometry Construction... circles-constructing-circle.jpg  
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    Why does this work though? What is the proof behind it?
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    Quote Originally Posted by MATNTRNG View Post
    Why does this work though? What is the proof behind it?
    If you draw numerous other purple circles of different sizes,
    you will see that the lines from their centres to the red line point of contact are all parallel.

    This is because if you extend the given lines out to infinity,
    when you draw the purple circle, it can be any size at all !!

    Imagine that you are a distance from the drawing....
    then you can magnify the drawing such that the purple circle is too small to see!
    Now roll the circle to the right and keep magnifying it.
    When it becomes visible, it will appear just as it had before.
    Attached Thumbnails Attached Thumbnails Geometry Construction... circles-constructing-circle-2.jpg  
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