Saturday, September 17, 2022

Geometry Problem 1497: Scalene Triangle, Transversal, Same Ratio, Proportionality, Isosceles, Double Angle

Geometry Problem 1497. Post your solution in the comment box below.
Level: Mathematics Education, K-12 School, Honors Geometry, College.

Details: Click on the figure below.

Geometry Problem 1497: Scalene Triangle, Transversal, Same Ratio, Proportionality, Isosceles, Double Angle

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6 comments:

  1. Where is Problem 1496?

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    1. Link: https://gogeometry.com/school-college/5/p1496-triangle-double-angle-isosceles-30-degree-congruence.htm

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  2. To Antonio : I published solution for this problem few weeks ago . The solution appear in few days then disappear later. I am not sure if there are something wrong in my solution ?

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    Replies
    1. Peter Tran has left a new comment on your post "Geometry Problem 1497: Scalene Triangle, Transversal, Same Ratio, Proportionality, Isosceles, Double Angle":

      https://photos.app.goo.gl/vwzCfb5NrWeMwChR6

      Problem 1497 solution 2
      Draw GK//HC
      C, E and K are collinear points => triangles HEC similar to GEK ( case AA)
      We have EH/EG=HC/GK=CE/CE => BG=GK
      Let BK meet DC at L
      Since EH/EG=CE/EK= CF/FB=> BK//EF ( Thales theorem)
      We have isosceles and similar triangles GBK and LAB ( case AA)
      And ∠ (ALB)= ∠ (ABL)= ∠ (FDC)=alpha
      And x= ∠ (HAG)= 2. Alpha

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  3. Peter Tran has left a new comment on your post "Geometry Problem 1497: Scalene Triangle, Transversal, Same Ratio, Proportionality, Isosceles, Double Angle":

    https://photos.app.goo.gl/LcWs7W3YV9TyGRYW8

    Draw BK//HC
    H, F and K are collinear points => triangles HFC similar to KFB ( Case AA)
    We have CF/BF=HF/FK=CH/BK=CH/BG => BG=BK
    Let GK meet DC at L
    Since EH/EG=CF/BF=HF/FK => GK//EF ( Thales theorem)
    We have isosceles triangles GBK and LAG
    And ∠ (ALG)= ∠ (AGL)= alpha
    And x= ∠ (HAG)= 2. alpha

    Posted by Peter Tran to Go Geometry (Problem Solutions) at Sep 17, 2022, 7:28 PM

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  4. My solution, given by the construction of the points E,F may be found at my blog : https://stanfulger.blogspot.com/2022/11/problem-1497-go-geometry.html
    Best regards,
    Stan Fulger

    ReplyDelete