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Math Help - Plane geometry problem

  1. #1
    Newbie
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    Sep 2012
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    Poland
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    Plane geometry problem

    Hey everyone,

    Recently, I've found an interesting task from plane geometry and I have no idea how to solve it

    Here it is (I translate it from my mother tongue, so sorry for any mistakes):
    There is a rhomboid ABCD with acute angle at A vertex.We suppose that circumcircle on triangle ABD intersects side CB in point K and side CD in point L (K and L are different from vertexes). Segment AN is a diameter of this circle. Prove that point N is a centre of circumcircle on triangle CKL.

    I add an image of this figure so that it's easier for you to understand the task:
    Plane geometry problem-om2.png
    Thanks for help in advance!
    If you have any questions (e.g. my translation is unclear), feel free to post them.
    Regards
    Lukasz
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  2. #2
    Super Member
    Joined
    Jun 2009
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    Re: Plane geometry problem

    Start by removing that line from B to D, it's distracting, but add lines from K and L to the centre of the big circle O.
    Call the angle at D \theta, then the angle AOL (the B side) will be 2\theta, (for angles on the same arc, the angle at the centre will be twice the angle at the circumference).
    That means the angle LON will be 2\theta - 180.
    Repeat the procedure for the other side of the figure and hence show that the angle KON is also 2\theta-180.
    It follows that the triangles LON and KON are congruent in which case LN will be the same length as KN.

    Since N is equidistant from L and K, it will lie on the perpendicular bisector of LK, which will be a diameter of the small circle.
    The base angles of the triangles LON and KON will be (180-(2\theta-180))/2=180-\theta,, in which case the angle LNK will be 360-2\theta.
    Finally, the angle at C is 180-\theta which is double the angle LNK. Putting this with the fact that N lies on a diameter, it follows that N is the centre of the small circle.
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  3. #3
    Newbie
    Joined
    Sep 2012
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    Poland
    Posts
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    Re: Plane geometry problem

    Exquisite proof!
    Thank you very much
    BTW- why don't you take a look at my second topic: Prove that n is square or doubled square
    Since you are so good, you may help me in solving it :P
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