Monday, May 19, 2008

Elearn Geometry Problem 72



See complete Problem 72
Intersecting Circles, Cyclic Quadrilateral, Angles, Parallel. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

4 comments:

  1. ABCD is cyclic so angle BAD = angle DCF. Similarly DCFE is cyclic and angle DCF = 180 -angle FED. Hence AB is parallel to EF.

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    Replies
    1. What about when BF intersects AE? that is another case, easy to proof, but is part of the problem.

      Greetings

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    2. Your case is just one of many varieties of configurations where Reim’s theorem can be applied straightforwardly.
      Enjoy!

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  2. This configuration is the general case illustrating Reim’s theorem.
    Anton Reim (1832–1922), czech mathematician.
    This theorem can easily be applied to some of the next problems proposed by Antonio Gutierrez: n° 73, 74, 75.
    The reciprocal il also true and is illustrated by the configuration shown in pb n° 77, where CD is a given chord in circle O, lines DEA and CFB with A and B such as AB//CD. The reciprocal of Reim’s theorem states that A,B,E,F are concyclic in circle O’ which intersects circle O in E and F.

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