Saturday, February 26, 2011

Problem 584: Cyclic Quadrilateral, Diagonals, Circumcenters, Circumcircles, Parallelograms

Geometry Problem
Click the figure below to see the complete problem 584.

Problem 584: Cyclic Quadrilateral, Diagonals, Circumcenters, Circumcircles, Parallelograms.
Zoom at: Geometry Problem 584

Posted by Antonio Gutierrez at 6:56 AM
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2 comments:

  1. c .t . e. oFebruary 27, 2011 at 6:49 AM

    EFGH from 583
    Name E', H', G' midpoint of AB, AD, CD
    AEM = 2ABM, DGM = 2MCD => AEM = DGM (1)
    From AEOH', DGOH' AEO = DGO (2)
    From (1) & (2) MEO = MGO (3)
    From AE'OH', DG'OH' EOG = A+D (4)
    ▲BEM => BME = 90 - BAM, ( BEM = 2BAM )
    ▲CMG => CMG = 90 - CDM
    ▲BMC => AMD = 180 - MAD - MDA
    => EMG = A+D (5)
    From (4) & (5) EOG = EMG (6)
    From (3) & (6) => EOGM parallelogram

    ReplyDelete
  2. Antonio GutierrezSeptember 3, 2012 at 2:30 PM

    Problem 584 solution: This solution was submitted by Michael Tsourakakis from Greece

    Thanks Mixalis.

    ReplyDelete

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