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Inequality with Circle and Triangle in Euclidean Geometry

  1. Nov 26, 2011 #1
    1. The problem statement, all variables and given/known data

    Please see below...

    2. Relevant equations

    Please see below...

    3. The attempt at a solution

    Hi. This question is on geometry with circle and triangle. I am stuck only on 2 parts of the solution and not the whole solution...

    Thank you....

    http://img256.imageshack.us/img256/9475/gtewp249no24.jpg [Broken]

    [ How did they get this? They never explained or proved this and it is NOT obvious from the picture.

    Blue: How is that implication true? By what theorem or reasoning?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Nov 29, 2011 #2
    To see that the red statement is true, observe that [itex]\angle[/itex]ADO = [itex]\angle[/itex]CDB > [itex]\angle[/itex]BOD by Euclid I.32 (book I proposition 32).

    In fact, that statement is superfluous. Once you know that [itex]\angle[/itex]ADO > [itex]\angle[/itex]BDO = [itex]\angle[/itex]ADC = [itex]\angle[/itex]AOD + [itex]\angle[/itex]OAD, you know that [itex]\angle[/itex]ADO > [itex]\angle[/itex]OAD, and by Euclid I.19, OA > OD.
     
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