Online Geometry theorems, problems, solutions, and related topics.
See complete Problem 85Contact Triangles Areas, Incircle, Excircle. Level: High School, SAT Prep, College geometryPost your solutions or ideas in the comments.
S area of triangle ABCS=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/rwith the results of pb83 and pb841/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.As 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.