Saturday, February 10, 2018

Geometry Problem 1355: Triangle, Midpoint, Median, Circle, Chord, Equal Product of measure of segments

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 1355: Triangle, Midpoint, Median, Circle, Chord, Equal Product of measure of segments.

4 comments:

  1. https://photos.app.goo.gl/h4PcQ74of1KJy3eh2
    Draw AP//DE and CN//DE
    See sketch for positions of points P,Q, N
    Triangle BAP is isosceles => BP=BA
    Triangles AQM and CNM are congruent ( case ASA) => AQ=NC
    Since DE//AP => GE/DG=PQ/AQ=PQ/NC
    Triangle BPQ similar to BCN => PQ/NC= BP/BC=AB/BC => GE/DG=AB/BC
    Or GE . BC= DG . AB

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  2. Draw perpendiculars h1 and h2 from G to AB and BC

    Since S (ABG) = S (CBG),
    AB.h1 = BC.h2..... (1)

    Since BD = BE,
    GE.h1 = DG.h2..... (2)

    Divide (1) by (2) and the result follows

    Sumith Peiris
    Moratuwa
    Sri Lanka

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  3. Using areas it is straight forward.
    Best regards

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  4. Let BM split angle ABC into angles x and y.
    Area of triangle ABM = Area of triangleBCM.
    So ½ AB.MB.sin x= ½ BC.MB.sin y,
    Implies AB/BC = sin y/sin x .
    Next EG/DG = (BGE)/(BGD)
    = ½ BE.BG sin y / ½ BD.BG sin x
    = sin y/sin x
    Hence AB/BC = EG/DG,
    AB.DG = BC.EG

    Vijaya Prasad Nalluri (Pravin)
    Rajahmundry - INDIA

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