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
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.
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.