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.

Posted by Antonio Gutierrez at 8:27 AM
Labels: , , , , ,

4 comments:

  1. Peter TranSeptember 7, 2011 at 6:36 PM

    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

    ReplyDelete
  2. PravinSeptember 8, 2011 at 3:57 AM

    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

    ReplyDelete
  3. BttfSeptember 25, 2011 at 10:20 PM

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

    chinese

    ReplyDelete
  4. Sumith PeirisFebruary 4, 2016 at 9:42 AM

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

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

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete

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