Sunday, September 7, 2008

Elearn Geometry Problem 174

Quadrilateral, Midpoint, Triangles, Area

See complete Problem 174 at:
www.gogeometry.com/problem/p174_quadrilateral_area_midpoint.htm

Quadrilateral with Midpoints, Triangles, Areas. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

3 comments:

  1. [BAF]+[EDC]=[BFD]+[BDE]=[BFE]+[EFD]=[EFC]+[EAF]=[EAFC]
    then S=S_1+S_2 is obvious

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  2. Let AD=2a, and the distances from B, E and C to AD be b, e and c respectively.
    Well known, e=(b+c)/2 ( 1 ) (trapezoid midline property). Multiply each side of (1) by a, to get ae=(ab+ac)/2, i.e. [AED]=[AFB]+[FCD], where from the required equality is obvious.

    Best regards

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  3. Define G the intersection of AE and BF and H the intersection of DE and CF
    Define M=[BGE], N=[EHC], P=[AGF], Q=[FHD], Sg=[GEF], Sh=[HEF] with Sg+Sh=S
    (1) : [ABCD]=[ABD]+[CBD]=2[ABF]+2[CDE]=2(S1+P+S2+N)
    P+Sg=Q+Sh and M+Sg=N+Sg => P-M=Q-N therefore (3) : P+N=Q+M
    (2) : [ABCD]=S1+S+S2+M+N+P+Q
    (1) and (2) : S1+S+S2+M+N+P+Q = 2(S1+P+S2+N)
    => S+M+Q= S1+P+S2+N
    But (3) : P+N=Q+M => S=S+S2

    ReplyDelete