Wednesday, September 7, 2011

Problem 665: Intersecting Circles, Secant, Tangent, Angles, Mind Map

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

Click the figure below to see the complete problem 665.

 Problem 665: Intersecting Circles, Secant, Tangent, Angles, Mind Map.

4 comments:

  1. http://img33.imageshack.us/img33/974/problem665.png
    Denote (XYZ)=angle(XYZ)
    CO interesect DO’ at F
    Note that (ECA)=(ABC)=beta
    And (ABD)=(ADE)=alpha
    Quadraliteral FCED is cyclic with (CFD)=(ECD)+(EDC)= alpha+beta=(CBD) (see picture)
    So 5 points F,C,E,D, B co-cyclic
    And ( EBD)=(ECD)= beta ( angles face the same arc DE)
    So x= (ABD)-(EBD)= alpha-beta
    Peter Tran

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  2. Join BC, BD
    ∠CBD = ∠ABD + ∠ABC = α + β
    ∠CED = 180° − (α + β)
    ∴B, C, E, D are concyclic
    So α = ∠CBE = x + ∠ABC = x + β
    ∴ x = α − β

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  3. http://imgsrc.baidu.com/forum/pic/item/38dbb6fd5266d0162ed21d77972bd40734fa35a1.jpg

    chinese

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  4. < CBA = beta and < ABD = alpha hence < CBD + CED = 180

    So < EBD = beta = alpha - x hence x = alpha - beta

    Sumith Peiris
    Moratuwa
    Sri Lanka

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