Geometry Problem. Post your solution in the comments box below.
Level: Mathematics Education, High School, Honors Geometry, College.
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Sunday, December 21, 2014
Geometry Problem 1068: Obtuse Triangle, Orthocenter, Circumradius, Inradius, Exradii, Distance, Diameter
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r = 4R sin(A/2) sin(B/2) sin(C/2)
ReplyDeleter₁ = 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.