Sunday, December 7, 2014

Geometry Problem 1067: Acute Triangle, Orthocenter, Circumradius, Inradius, Exradii, Distance, Diameter

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

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Online Math: Geometry Problem 1067: Acute Triangle, Orthocenter, Circumradius, Inradius, Exradii, Distance, Diameter.
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1 comment:

  1. Jacob HA (EMK2000)December 7, 2014 at 4:47 PM

    r = 4R sin(A/2) sin(B/2) sin(C/2)
    r₁ = 4R sin(A/2) cos(B/2) cos(C/2)
    r₂ = 4R cos(A/2) sin(B/2) cos(C/2)
    r₃ = 4R cos(A/2) cos(B/2) sin(C/2)

    a₁ = 2R cosA
    b₁ = 2R cosB
    c₁ = 2R cosC

    r₁ + r₂ + r₃ − r = 4R
    r₁ + r₂ + r₃ + r = 4R + 2r

    a₁ + b₁ + c₁ + 2R = 2R (cosA + cosB + cosC + 1)
    = 2R (2 + 4 sin(A/2) sin(B/2) sin(C/2))
    = 4R + 2r

    Hence, r₁ + r₂ + r₃ + r = a₁ + b₁ + c₁ + 2R.

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