Friday, November 4, 2016

Geometry Problem 1283 Two Equilateral Triangle, Perpendicular, Midpoint

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

Click the figure below to view more details of problem 1283.

Geometry Problem 1283: Two Equilateral Triangle, Perpendicular, Midpoint.
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3 comments:

  1. Peter TranNovember 4, 2016 at 4:03 PM

    https://goo.gl/photos/96rSC1H87PN4Xb7g7

    Let N is the projection of M over CDE
    In triangle EMD, MN is the median and altitude from M => MED is isosceles triangle
    Connect CM , CM ⊥AB and BE⊥EC => quadri. CMBE is cyclic
    in cyclic quadrilateral CMBE ∠MDC=∠MBC=60 degrees => Triangle EMC is equilateral

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  2. APOSTOLIS MANOLOUDISNovember 4, 2016 at 11:21 PM

    Problem 1283
    Suppose that MN is perpendicular to CE ( DN =NE ,ABED is trapezoid ) so triangle MED is isosceles (ME=MD).But <AMC=90=<ADC then AMDC is cyclic. So <MDA=<MCA=60/2=30.
    Then <MDE=90-30=60.Therefore triangle MDE is equilateral.
    APOSTOLIS MANOLOUDIS 4 HIGH SCHOOL KORYDALLOS PIRAEUS GREECE

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  3. Sumith PeirisNovember 5, 2016 at 7:20 AM

    BECM is cyclic with BC as diameter so
    < MEC = < MBC = 60

    ACDM is cyclic with AC as diameter so
    < MDE < MAC = 60

    So 2 angles of Tr. MDE = 60 and is hence equilateral

    Sumith Peiris
    Moratuwa
    Sri Lanka

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