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Math Help - Inscribe a circle into a sector

  1. #1
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    Inscribe a circle into a sector

    The question is "Inscribe a circle into the sector below."
    It can be any sector, and the circle has to be of maximum area, how would I do this?
    Sector = Sector of a circle
    Last edited by arrowhead566; March 7th 2010 at 02:02 PM.
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  2. #2
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    Hi arrowhead566,

    if we take a sector of a circle,
    the largest "incircle" of this must be tangential to the major circle.
    It's centre must be on the bisector of the sector's angle,
    hence, it's centre is the incentre of the resultant triangle
    shown in the attachment.
    Attached Thumbnails Attached Thumbnails Inscribe a circle into a sector-sector-incircle.jpg  
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  3. #3
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    How would you construct the actual larger circle, and what do you mean by incentre?
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  4. #4
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    If you only have the sector,
    there is no need to construct the larger circle.
    You only need the triangle.
    Just bisect the pointed angle of the sector,
    draw the bisector and locate where this bisector intersects the sector arc.
    This is where the 3rd side of the isosceles triangle is tangential to the sector arc.
    Therefore that side can be constructed.

    Now that you have the triangle, the circle centre is the triangle's incentre.
    This incentre is equidistant from the 3 sides.
    Hence, to find this you must bisect the 3 angles of the triangle.
    One is already done, so now bisect one of the other 2.
    The point of intersection of the corner angle bisectors is the circle centre.
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