Friday, September 9, 2011

Problem 667: Intersecting Circles, Secant, Tangent, Metric Relations, Mind Map

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

Click the figure below to see the complete problem 667.

Problem 667: Intersecting Circles, Secant, Tangent, Metric Relations, Mind Map.

Posted by Antonio Gutierrez at 4:03 PM
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2 comments:

  1. http://img716.imageshack.us/img716/570/problem667.png
    Per the result of problem 666, quadrilateral CEDB is cyclic
    So (ECD)=(EBD)= β = (EDF)
    And (CBF)=(CDE)= α
    In triangle EGD we have (CGD)=(GDE)+(GED)
    = β+ ( 180- α- β)=180- α
    So (CGF) supplement to (CBF) and quadrilateral CGFB is cyclic
    We have EF.EB=EG.EC= ED^2
    And x^2=4 x 9= 36 ; x=6
    Peter Tran

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  2. Let EB intersect circle (O) again at H.
    Join CB, AB
    ∠BCG
    = ∠BCA + ∠ACG
    = ∠BCA + ∠CBA
    = ∠BAD
    = ∠BFD
    = ∠GFE
    So G, C, B, F are concyclic
    Hence x² = EF.EB = EG.EC = 4 x 9 = 36
    ∴ x = 6

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