See complete Problem 209 at:
gogeometry.com/problem/p209_triangle_incircles_inradius.htm
Triangle, Incircles, Inradius, Contact triangle. Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
Friday, November 21, 2008
Elearn Geometry Problem 209: Triangle, Incircles, Inradius, Contact triangle
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1)triangles AOF and AOD are congruent
ReplyDeletethe bisector of angle DAF is also the bisector of the central angle DOF,M midpoint of arc DF lies on this bisector
M lies on the bisector of inscribed angle AFD whose leg AF is tangent to the incercle
the bisectors of triangle ADF meet at M=O1
2)in triangle AOF
sin(A/2)=r/AO=r'/(AO-r)
r'=r(1-sin(A/2))
the same way
r''=r(1-sin(B/2);r'''=r(1-sin(C/2))
in triangle DEF,r is the circumradius and r4 is the inradius
r4=4rsin((A+B)/4)sin((B+C)/4)sin((C+A)/4)
after substitution and appropriate transformation
r'+r''+r'''+r4=2r
.-.
About the solution of problem 209 by Anonymous - December 31, 2009:
ReplyDeleteHow can I get the result
r4 = 4r.sin((A+B)/4).sin((B+C)/4).sin((C+A)/4)?
Can anyone explain it to me, please?
Thank you.