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Math Help - Bisecting a triangle with four chords of a circle

  1. #1
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    Smile Bisecting a triangle with four chords of a circle

    There are 4 chords in a circle. NK, NL, NM and KM.
    C is the point in the center of the chord KM.
    Also, the radius from the circle's origo O to point L is a line which passes through C and this line OL is perpendicular to chord KM.

    Question:
    Is chord NL always bisecting triangle KNM?
    I.e., is angle KNL always = angle MNL?
    Regardless of where N is on the circle? Given that all other points are fixed and it is "above" the chord KM as in the figure.

    Bisecting a triangle with four chords of a circle-4chords1bisect.png
    Methinks it should be, but I fail to derive it properly why it must. So help is asked!

    Best regards
    Last edited by Encircled; January 28th 2013 at 01:47 PM.
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  2. #2
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    Athens, OH, USA
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    Re: Bisecting a triangle with four chords of a circle

    Yes, it's true. The Inscribed Angle Theorem states that for any chord, the inscribed angle is 1/2 of the central angle. For example, in your diagram, angle KNL is 1/2 angle KOL. In your case angle KNL = 1/2 angle KOL = 1/4 angle KOM.
    Also angle MNL = 1/2 angle MOL = 1/4 angle KOM.

    The Inscribed Angle Theorem is a good theorem to add to your arsenal. A good discussion is found at wikapedia. This theorem is in Euclid's Elements.
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