Friday, September 23, 2011

Problem 672: Internally tangent circles, Chord, Tangent, Geometric Mean

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 672.

Online Geometry Problem 672: Internally tangent circles, Chord, Tangent, Geometric Mean.

5 comments:

  1. Extend AD to meet circle O again at E
    Circles O and O' touch each other at A
    So points A, O', O are collinear
    OA being a diameter(of circle O'),
    ODA is a right angle
    So OD bisects the chord ADE in circle O
    Now a.b = AD.DB = AD.DE = AD^2
    Hence AD = sqrt(a.b)

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  2. Extend AD to meet circle O at E. Since OA is a diameter of circle O',OD is perpendicular to AD or OD is perpendicular chord AE of circle O. In other words D is the midpt. of AE which means AD*DC = AD*DE = AD^2 (intersecting chords) or a*b = AD^2 or AD is the GM of a & b.
    Ajit : ajitathle@gmail.com

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  3. Typo Correction:
    BD * DC = AD * DE = AD^2 since AD = DE
    Ajit

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  4. Chinese version
    http://imgsrc.baidu.com/forum/pic/item/500fd9f9d72a6059e27a5d452834349b033bba03.jpg

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  5. Let AD extended meet circle O at P

    < ODA = 90 so PD = DA = x

    Hence easily in circle O,

    PD. DA = a.b i.e. x^2 = a.b

    Sumith Peiris
    Moratuwa
    Sri Lanka

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