Friday, May 31, 2019

Geometry Problem 1437: Intersecting Circles, Common Tangent, Equal Product

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 1437: Intersecting Circles, Common Tangent, Equal Product, Poster, iPad apps, Tutoring.


  1. O1O2 ∩ CD=K
    we 'll prove E,F,K in the same line (sorry for my english)
    K ∈ Δ ,Δ ∩ (o2)=g,Δ ∩ (o1)=h
    so KD/KC=KG/KH and so DG // EH

    Back to our problem we have that CDEF is Inscribed quadrilateral ,so DCF=DEF
    EF∩(O1)=M , DCF=CMF so CMF=DEF so DE//CM
    we can see with any Δ (K∈Δ) we'll have DG//CH so if DE//CM we'll have E and M ∈ Δ or
    K,F,E,M ∈ Δ

    KA ∩ (c2)=N,KB ∩ (c2)=P
    we have AFMN is Inscribed quadrilateral so ΔKFA congruent with ΔKNM
    so AF/MN=KA/KM(1)
    just like that BF/MP=KB/KM(2)
    (1):(2) so AF/BF=(KA/KB)*(MN/MP)(3)
    and o2o1 perpendicular with AB ,K ∈ o2o1 so KA=KB (4)
    from (3) and (4) we'll have AF/BF=MN/MP(a)

    we can see that DA//CN,DE//CM so KA/KN=KE/KM or AE//MN (5)
    just like that we also have BE//MP (6)
    from (5) and (6) we'll have AE//BE=MN/MP(b)

    And from (a) and (b) we'll have what we need

  2. Bro,try to give a figure if the solution is very much constructive.