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Grade: 11

                        

in a triangle ABC a-b/b-c=s-a/s-c then exradii r1,r2,r3 are in a)AP b)GP c)HP

2 years ago

Answers : (1)

mycroft holmes
272 Points
							
You have \frac{s-a}{a-b} = \frac{s-c}{b-c}
 
Subtracting 1 from both sides you get \frac{s-b}{a-b} = \frac{s-b}{b-c} \Rightarrow a-b = b-c
 
Hence a,b,c are in AP, and so s-a, s-b, s-c are in AP and hence
 
\frac{s-a}{\Delta} , \frac{s-b}{\Delta} , \frac{s-c}{\Delta} are in AP
 
i.e. \frac{1}{r_1} , \frac{1}{r_2} ,\frac{1}{r_3} are in AP
 
and hence the ex-radii are in HP
2 years ago
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