Grade 11Algebra

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

Mitanshi

8 Years agoGrade 11

Approved Tutor Answer8 Years ago

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

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