Thursday, November 6, 2014

Geometry Problem 1056: Triangle, Exradius, Reciprocals of the Altitudes, Multiplicative Inverse, Perpendicular, Excircle, Circle

Geometry Problem. Post your solution in the comments box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the diagram below to enlarge it.

Online Math: Geometry Problem 1056: Triangle, Exradius, Reciprocals of the Altitudes, Multiplicative Inverse, Perpendicular, Excircle, Circle.

2 comments:

  1. Use subscript 1,2,3 instead of a,b,c.
    Let BC=a, AC=b, AB=c. Let Δ be the area of ABC.

    r₁ = Δ/(s−a), 1/r₁ = (s−a)/Δ = (b+c−a)/(2Δ)
    r₂ = Δ/(s−b), 1/r₂ = (s−b)/Δ = (a+c−b)/(2Δ)
    r₃ = Δ/(s−c), 1/r₃ = (s−c)/Δ = (a+b−c)/(2Δ)

    ah₁ = 2Δ, 1/h₁ = a/(2Δ)
    bh₂ = 2Δ, 1/h₂ = b/(2Δ)
    ch₃ = 2Δ, 1/h₃ = c/(2Δ)

    The rest is obvious.

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  2. Let S= area of triangle ABC= Area(ACE)+Area(ABE)-Area(BCE)
    S= ½(.Ra).b+1/2(Ra).c-1/2(Ra).a= ½.(Ra).(b+c-a)
    So 1/(Ra)=1/2(b+c-a)/S ..... (1)
    But b/S=2/(Hb) and c/S=2/(Hc) , a/S=2/(Ha)
    Replace it in (1) we have 1/(Ra)=1/(Hb)+1/(Hc)-1/(Ha)

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