See complete Problem 85
Contact Triangles Areas, Incircle, Excircle. Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
Monday, May 19, 2008
Geometry Problem 85
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See complete Problem 85
Contact Triangles Areas, Incircle, Excircle. Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
S area of triangle ABC
ReplyDeleteS=pr=r_a(p-a)=r_b(p-b)=r_c(p-c)
1/r_a+1/r_b+1/r_c=(p-a+p-b+p-c)/S=p/(pr)=1/r
with the results of pb83 and pb84
1/S_a+1/S_b+1/S_c=2R/S(1/r_a+1/r_b+1/r_c)=
2R/(rS)=1/S_1
.-.
Here is a slightly different solution to problem 85.
ReplyDeleteAs the area of triangle ABC is S = pr = (p-a).ra = (p-b).rb = (p-c).rc, then 1/ra + 1/rb + 1/rc = (1/pr).(p-a+p-b+p-c)=(1/pr).(3p-2p)=1/r.
From the results of problem 84 we have S1/Sa + S1/Sb + S1/Sc = r(1/ra + 1/rb + 1/rc) = 1, so 1/Sa + 1/Sb + 1/ Sc = 1/S.
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ReplyDelete