In-a-ABC-if-s-a-a-b-s-c-b-c-then-prove-that-r-1-r-2-r-3-are-in-A-P-Here-r-1-r-2-and-r-3-are-the-exradii-opposite-to-angles-A-B-and-C-respectively-

Question Number 16359 by Tinkutara last updated on 21/Jun/17

In a Δ A B C if \frac{s - a}{a - b} = \frac{s - c}{b - c}, then

prove that r_{1}, r_{2}, r_{3} are in A . P .

Here r_{1}, r_{2} and r_{3} are the exradii

opposite to angles A, B and C respectively .

Answered by ajfour last updated on 21/Jun/17

r_{1} = \frac{Δ}{s - a}, r_{2} = \frac{Δ}{s - b}, r_{3} = \frac{Δ}{s - c}

G i v e n \frac{s - a}{(s - b) - (s - a)} = \frac{s - c}{(s - c) - (s - b)}

o r \frac{(s - b) - (s - a)}{s - a} = \frac{(s - c) - (s - b)}{s - c}

\frac{r_{1}}{r_{2}} - 1 = 1 - \frac{r_{3}}{r_{2}}

o r 2 r_{2} = r_{1} + r_{3}

\Rightarrow r_{1}, r_{2}, a n d r_{3} a r e i n A . P .

Commented by Tinkutara last updated on 21/Jun/17

Thanks Sir!