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Math Help - Geometry Construction Circles II

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    Geometry Construction Circles II

    Given two circles, 1 and 2, with centers at A and B. Let E be one of the two points where the circles intersect. Construct a line FG, through E, on which the two circles cut out equal chords, where neither F nor G are at the other intersection point of the two circles.
    Geometry Construction Circles II-two-circles.jpg
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    Quote Originally Posted by MATNTRNG View Post
    [FONT=Times New Roman][SIZE=3]Given two circles, 1 and 2, with centers at A and B. Let E be one of the two points where the circles intersect. Construct a line FG, through E, on which the two circles cut out equal chords, where neither F nor G are at the other intersection point of the two circles.
    1. Construct the midpoint M of AB.
    2. The line perpendicular to ME through E yields the 2 chords with equal length. (see attachment)
    3. Prove that this construction is valid.
    Attached Thumbnails Attached Thumbnails Geometry Construction Circles II-glstrckinkreisen.png  
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    Quote Originally Posted by earboth View Post
    3. Prove that this construction is valid.
    There is no doubt in my mind that this construction is valid. How would I go about proving it?
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    Quote Originally Posted by MATNTRNG View Post
    There is no doubt in my mind that this construction is valid. How would I go about proving it?
    Hi MATNTRNG,

    One way is to show that if we draw 2 more circles,
    same radii as Circle1 and Circle 2, but centred on the original circle's centres as shown,
    the geometry is entirely symmetrical since triangle FmG is isosceles,
    due to the two chords being of equal length.

    We can repeat the geometry under the centreline,
    hence point "m" is on the centreline,
    and due to the left-right symmetry, also must be midway between the circle centres.
    Attached Thumbnails Attached Thumbnails Geometry Construction Circles II-equal-chords.jpg  
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    Quote Originally Posted by MATNTRNG View Post
    There is no doubt in my mind that this construction is valid. How would I go about proving it?
    The previous post is really not suitable as a proof, MATNTRNG.

    Instead, the attachment shows that if the chords are to be of equal length,
    then they form the basis of numerous isosceles triangles, two of which are shown.

    Since the chords themselves form isosceles triangles with the circle centres, then

    [m_1o_1] and [m_2o_2] are parallel.

    These line segments are also parallel to the perpendicular line passing through
    the point at which the chords meet.
    The point on the joint-centreline of the two circles that is equidistant from F and G lies on that perpendicular line
    and therefore is midway between the centres.
    Attached Thumbnails Attached Thumbnails Geometry Construction Circles II-equal-chords2.jpg  
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