Friday, March 21, 2014

Geometry Problem 994: Trapezoid, Midpoints of the bases, Concurrent Lines

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

Click the figure below to view the complete problem 994

Online Geometry Problem 994: Trapezoid, Midpoints of the bases, Concurrent Lines.

3 comments:

  1. Let CD cut MN at P
    ∆ PNC similar to ∆PMD
    So PN/PM=NC/MD=NB/MA => ∆ PNB similar to ∆ PMA
    So A, B, P are collinear and AB, MN, DC concurrent at P

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  2. Problem 994
    Is BN/AM=NC/MD then from the inverse theorem batch AB,MN and DC are concurrent at P.
    APOSTOLIS MANOLOUDIS 4 HIGH SCHOOL KORYDALLOS PIRAEUS GREECE

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  3. M is the middle AD, P is the intersection of AB and CD, N is the intersection of BC and MP
    We have to demonstrate that N is the middle of BC
    AD//BC => ∠MAP=∠NBP and ∠MDP=∠NCP
    => Δ MAP is similar to Δ NBP (AA)
    => (1) PN/NB = PM/MA
    => Δ MDP is similar to Δ NCP (AA)
    => (2) PM/MD = PN/NC
    MA=MD => (3) PM/MA = PM/MD
    (1&3&2) PN/NB = PM/MA = PM/MD = PN/NC
    => PN/NB= PN/NC => NB=NC
    => N is the middle of BC

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