Friday, March 22, 2019

Geometry Problem 1427: Circle inscribed in a square, Arc, Tangent Line, Tangent Circles, Radii

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

Geometry Problem 1427: Circle inscribed in a square, Arc, Tangent Line, Tangent Circles, Radii, Tutoring.

Posted by Antonio Gutierrez at 9:02 PM
Labels: , , , , ,

5 comments:

  1. Sumith PeirisMarch 22, 2019 at 11:42 PM

    Let the square be of side 2a.
    Denote sqrt2 by p.

    Then pr1 + r1 = 2p.a - 2a
    So r1 = 2(p-1)/(p+1)a.....(1)

    Also p.r2 + r2 + a = p.a
    Hence r2 = (p-1)/(p+1)a ....(2)

    From (1) and (2)
    r1 = 2.r2

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete
  2. Peter TranMarch 23, 2019 at 8:08 AM

    https://photos.app.goo.gl/bzbDb5w5sQ24aZnf6

    Let AB=AD=2.a
    We have CE=AC-AE=2a.(sqrt(2)-1)= O1E+O1C= r1(1+sqrt(2))
    So r1= 2a(sqrt(2)-1) ^2
    Similarly we also have r2= a(sqrt(2)-1) ^2
    So r1= 2 r2

    ReplyDelete
    Replies
    1. Peter TranOctober 22, 2020 at 11:34 AM

      https://photos.app.goo.gl/V4sKjMDJsFrQDuMBA

      New approach to solve this problem
      Let O3 is the circle shown on the sketch
      Obviously r3=r2
      perform homothety transformation center C, factor= 2.
      Circle O will become circle A radius AB. Lines BC and CD will be the same
      O3 become circle O1
      So r1=2r3=2r2

      Delete
  3. c.t.e.oMarch 23, 2019 at 8:30 AM

    From AO => R(√2-1) = r2(√2+1)
    From AC => 2R(√2-1) = r1(√2+1)

    ReplyDelete
  4. rv.littlemanJuly 5, 2021 at 2:29 PM

    r2 is the radius of a circle tangent to a square with side D and its incircle
    then 2r2 is the radius of a circle tangent to a square with side 2D and its incircle
    Therefore r1=2r2

    ReplyDelete

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