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Affine Geometric Problem

  1. May 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the ratio of areas is proportional to the sides squared:

    [tex]\frac{[ACD]}_{[CDB]}[/tex] is proportional to [tex] \frac{AC^2}_{CB^2} [/tex]

    Please, see the picture: http://dl.getdropbox.com/u/175564/geo_henry.JPG [Broken].

    2. Relevant equations

    AC = 2 * CB

    [tex] \frac{AD}_{DB}[/tex] is propotional to [tex]\frac{[ACD]}_{[CDB]} [/tex]

    where [ACD] and [CDB] are areas.

    3. The attempt at a solution

    I was unable to prove the relation with pythagoras, so I feel an easier solution. Perhaps, you can prove it somehow with affine geometry.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. May 11, 2009 #2
    Picture

    Please, see the picture here: http://dl.getdropbox.com/u/175564/geo_henry.JPG [Broken]. Or download the attachment.
     

    Attached Files:

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    Last edited by a moderator: May 4, 2017
  4. May 11, 2009 #3

    Mark44

    Staff: Mentor

    The basic idea is that if two triangles are similar (same corresponding angles), their corresponding sides will be proportional.

    [itex]\angle[/itex]ACD = [itex]\angle[/itex]CDM - alt. interior angles cut by transversal
    [itex]\angle[/itex]MDB = [itex]\angle[/itex]DAC - complements of congruent angles are congruent
    [itex]\angle[/itex]ABC = [itex]\angle[/itex]ACD - complements of congruent angles are congruent

    The statements above show that triangle ACD is similar to triangle CDB.
    [ACD] = 1/2 * AD * CD
    [CDB] = 1/2 * DB * CD

    The two equations above show the proportionality you want.
     
  5. May 12, 2009 #4
    I am sorry of the blurry image. I wanted to know why the relation is true:

    [tex] \frac{AC^2}_{CB^2} [/tex] is proportional to [tex]\frac{[ACD]}_{[CDB]}[/tex]
     
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