Wednesday, May 30, 2012

Problem 758: Equilateral Triangle, Cevians, Trisection of sides, Congruence, Perpendicular, 90 Degrees

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

This entry contributed by Ajit Athle.
Click the figure below to see the problem 758 details.

Online Geometry Problem 758: Equilateral Triangle, Cevians, Trisection of sides, Congruence, Perpendicular, 90 Degrees.

6 comments:

  1. Let vec(BA)=a, vec(BC)=b.

    By Menelaus' theorem in triangle BEC and line AFD,
    EF/FC * CD/DB * BA/AE = EF/FC * 2/1 * 3/1 = 1
    so EF/FC = 1/6.

    vec(BE) = 2/3 a
    vec(BF) = [6 vec(BE) + vec(BC)]/7 = (4a+b)/7
    vec(CE) = vec(BE) - vec(BC) = (2a-3b)/3

    Now let |a|=|b|=k, then
    a.a = k^2
    b.b = k^2
    a.b = k*k*cos(60) = k^2 / 2

    Therefore,
    vec(BF).vec(CE) = (4a+b).(2a-3b)/9
    = [8 a.a - 10 a.b - 3 b.b] / 9
    = [8 - 5 - 3]k^2 / 9
    = 0

    Hence, BF perp CE.

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  2. Haven't found the synthetic solution yet but here's my analytical attempt: WLOG assume the midpoint of AC as the origin (0,0) and let A:(-1.0) & B:(1.0) hence C:(0,√3). We can now determine D to be (1/3,2/√3) & E:(-2/3,1/√3). Slope CE=-√3 /5 and
    eqn CE: 5y =-√3x + √3. Similarly, slope AD = √3/2 and eqn. AD: 2y =√3x + √3. Solve the two eqns. simultaneously to obtain F as (-3/7,2√3/7). Therefore, slope BF = 5/√3 and finally (slope BF * slope CE)= -1. Hence BF _l_ CE.
    Will keep looking for a synthetic solution.

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  3. http://henrik-geometry.webs.com/758.htm

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  4. ∆ABD≡∆CAE
    ∴ ∠AEC = ∠ADB and so B,E,F,D are concyclic
    ∴ ∠AFE = 60°
    Let M be the midpoint of BE
    ∆BMD is equilateral
    Since ME = MB = MD, ∆EDB is right angled at D
    B,D,E,F being concyclic,
    we have ∠BFE = ∠BDE = 90°

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  5. Tr.s CAE & ABD are congruent. So Tr.s AEF & ABD are similar since Tr.s AEF & CAE are similar. Hence BEFD is con cyclic.
    Now consider Tr. BDE where BE = 2. BD & the included < is 60. So Tr. BDE is a 30-60-90 Tr. and the result follows.

    Sumith Peiris
    Moratuwa
    Sri Lanka

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