Monday, May 19, 2008

Elearn Geometry Problem 32: Triangle and Quadrilateral

Triangle, Cevian, Incircles

See complete Problem 32 at:
www.gogeometry.com/problem/p032_triangle_incircle_tangent.htm

Triangle, Cevian, Incircles. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

3 comments:

  1. Hint: Apply
    Two tangent theorem:
    Two tangent segments to a circle from an external point are congruent.

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  2. See the drawing

    GN=GK=rg, DK and DN tangent to CircleG => DK=DN
    FL=FM=rf, DL and DM tangent to CircleF => DL=DM
    F center of circleF => F is on the angle bisector of ∠ADB => ∠FDA= ∠FDB=x
    G center of circleG => G is on the angle bisector of ∠BDC => ∠GDB= ∠GDC=y
    ∠ ADC= π =2x+2y => x+y= π/2 => ∠GDF=π/2
    =>DG ⊥ DF
    DM=DL and FL=FM => DMFL is a kite=>ML ⊥ FD
    Draw a line parallel to AB and passing through G
    Define Z the intersection of this line and HE
    ML ⊥ FD, Z on ML => MZ ⊥ FD
    DG ⊥ FD => MZ//DG => ∠ZMH = ∠GDN
    ∠ZMH = ∠GDN and ∠MHZ = ∠DNG and HZ=NG =>ΔMHZ is congruent to ΔDNG
    = >MH=DN
    DL=DM => DK+KL=DH+HM =>DN+KL=DH+DN
    Therefore LK=HD


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