Tuesday, August 5, 2008

Elearn Geometry Problem 157



See complete Problem 157
Distance from the Circumcenter to the Excenter. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

3 comments:

  1. join A to E, A to O
    draw OG perpendicular to AE

    OG² = R² - (AD/2)² ( from tr AOG )
    OG² = d² - (AD/2 + DE)² ( from tr OGE )

    R² - (AD/2)² = d² - (AD/2)² - AD∙DE - DE²
    R² = d² - DE∙( AD + DE )
    d² = R² + DE∙AE

    from P156 => DE∙AE = 2Rr

    d² = R² + 2Rr
    ------------------------------------------

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  2. http://ahmetelmas.wordpress.com/2010/05/15/geo-geo/

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  3. Solution to problem 157.
    Line AE meets the circumcircle at D. Segment EO meets the circumcircle at F, and the extension of EO meets the circumcircle at G.
    Taking the circle power of point E with relation to the circumcircle we have EF.EG = ED.EA. But EF.EG = (d-R)(d+R) = d^2 – R^2. By problem 156 we know that ED.EA = 2.R.r1. Hence d^2 – R^2 = 2.R.r1, and d^2 = R^2 + 2.R.r1.

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