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

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

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## Thursday, December 4, 2014

### Geometry Problem 1066: Triangle, Obtuse Angle, Orthocenter, Circumradius, Inradius, Exradius, Distance, Diameter

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

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Since ΔABC and ΔHBC have the same circumradius,

ReplyDeletea₁ = 2R cosH = −2R cosA

r = 4R sin(A/2) sin(B/2) sin(C/2)

r₁ = 4R sin(A/2) cos(B/2) cos(C/2)

r₁ − r = 4R sin(A/2) cos(B/2 + C/2)

= 4R sin²(A/2) = 2R(1−cosA) = 2R + a₁

a₁ + r₁ − r = 2R

r₁ − a₁ = 2R + r