Geometry Problem. Post your solution in the comments box below.

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

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## Sunday, December 7, 2014

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

Labels:
acute,
circumradius,
diameter,
distance,
exradius,
inradius,
orthocenter,
triangle

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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.