Thursday, January 7, 2016

Geometry Problem 1175: Six Tangential or Circumscribed Quadrilaterals

Geometry Problem. Post your solution in the comments box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to view more details of problem 1175.

Online Math: Geometry Problem 1175: Six Tangential or Circumscribed Quadrilaterals .

Posted by Antonio Gutierrez at 11:51 AM
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4 comments:

  1. Peter TranJanuary 7, 2016 at 7:26 PM

    This problem is almost identical to problem 883.
    see link below for the solution.

    http://gogeometry.blogspot.com/2013/06/problem-883-five-tangential-or.html

    Peter Tran

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  2. Sumith PeirisJanuary 7, 2016 at 10:55 PM


    BK must bisect angles < B and < K. KG similarly < K and < G and GD < G and D. Hence BKGD is collinear and bisects < B and < D

    Similarly AFLC can be shown to be collinear bisecting < A and < C

    So the diagonals of ABCD AC and BD bisect the 4 angles and hence ABCD must be a tangential quadrilateral

    Sumith Peiris
    Moratuwa
    Sri Lanka

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  3. Peter TranJanuary 9, 2016 at 7:44 PM

    To Sumith Peiris

    Refer to your solution
    1. In general case diagonal BK of tangential quadrilateral BNKJ is not bisect angle B or angle D . Please justify for statement.
    2. In my opinion B, K, G, D are not collinear as per your solution. Please justify

    Peter Tran

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  4. Sumith PeirisJanuary 10, 2016 at 10:54 AM

    I get your point Peter.

    In a quadrilateral if the diagonals bisect the angles the quadrilateral is easily shown to be tangential with the point of intersection of the diagonals the centre of this in circle

    The converse is not necessarily true

    If for example U is the centre of circle within BNKJ, V of KLGF and W of GHDR,then

    UKV and VGW are collinear but BUK and GWD need not necessarily be so.

    Hence my proof is fallacious

    Thanks Peter

    Antonio /Peter - any ideas as to how my proof could be corrected?

    ReplyDelete

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