Tuesday, January 20, 2009

Elearn Geometry Problem 228: Triangle, Midpoint, Exterior line

Triangle, Midpoints, Exterior line

See complete Problem 228 at:
gogeometry.com/problem/p228_triangle_midpoint_perpendicular.htm

Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.
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2 comments:

  1. macsim gabriel florinFebruary 2, 2009 at 7:43 AM

    We have:
    in the trapezoid AA'B'B : d=(a+b)/2
    in the trapezoid BB'C'C : e=(b+c)/2
    in the trapezoid AA'C'C : f=(a+c)/2
    From this relations we have :
    d+e+f=(2a+2b+2c)/2=a+b+c

    ReplyDelete
  2. rv.littlemanSeptember 13, 2020 at 12:28 PM

    See the drawing

    - DD’ intersects AC in D’’, BB’ in B’’, EE’ in E’’
    - Define d1=DD’’, d2=D’’D’ with d=d1+d2
    - Define b1=BB’’, b2=B’’B’ with b=b1+b2
    - Define e1=EE’’, e2=E’’E’ with e=e1+e2
    - F in the middle of AC =>FF’=f=(a+c)/2
    - DAD’’ is similar to BAB’’ =>BA/DA=b1/d1
    - BA/DA=2=> d1=b1/2
    - In the same way, e1=b1/2=d1
    - DAD’’ is similar to BAB’’ =>BA/DA=B’’A/D’’A=2 => D’’ A=D’’B’’
    - In the same way, E’’B’’=E’’C
    - =>d2=(a+b2)/2 and e2=(c+b2)/2
    - d+f+e=d1+d2+f+e1+e2=b1/2+(a+b2)/2+(a+c)/2+(b1/2)+(c+b2)/2
    - =(b1+a+b2+a+c+b1+c+b2)/2=(2b+2a+2c)/2
    - =>d+f+e=a+b+c

    ReplyDelete

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