Wednesday, November 13, 2019

Dynamic Geometry Problem 1446: The Lemoine Line

Interactive step-by-step animation using GeoGebra. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

Dynamic Geometry Problem 1446: The Lemoine Line. Using GeoGebra.


    Let circle center E, radius=EA cut BC at M and L ( see sketch)
    Note that ∠ (ABC)= ∠ (EAC)=u and triangle EAM is isosceles
    ∠ (BAM)= ∠ (AMC)-u= ∠ (MAC) => AM and AL are angle bisectors of angle BAC
    Calculate MC= a.k/(1+k) , MB=a/(1+k)
    LC=a.k/(1-k) ; LB=a/(1-k)
    And EC= ak^2/(1-k^2) ; EB= a/(1-k^2) ; EC/EB= k^2= b^2/c^2
    Similarly we have DB/DA= a^2/b^2 and FA/FC= c^2/a^2
    Verify that EC/EB x DB/DA x FA/FC= 1 => E,D,F are collinear per Menelaus’s theorem

  2. Sorry Peter Tran, your sketch at can not be open. Thanks