In triangle ABC, if 1−r1r21-r1r3=2, then the triangle is

1-r1r21-r1r3=21-s-bs-a1-s-cs-a=2b-as-ac-as-a=2b-cc-a=2s-a2⇒b2+c2=a2The triangle is right angled

In triangle ABC, if 1−r1r21-r1r3=2, then the triangle is

1-r1r21-r1r3=21-s-bs-a1-s-cs-a=2b-as-ac-as-a=2b-cc-a=2s-a2⇒b2+c2=a2The triangle is right angled

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In triangle ABC, if $(1 - \frac{r_{1}}{r_{2}}) (1- \frac{r_{1}}{r_{3}}) = 2$ , then the triangle is

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right angled

scalene

Isosceles

Equilateral

answer is C.

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Detailed Solution

$(1- \frac{r_{1}}{r_{2}}) (1- \frac{r_{1}}{r_{3}}) =2$
$(1 - \frac{s - b}{s - a}) (1 - \frac{s - c}{s - a}) = 2$
$(\frac{b - a}{s - a}) (\frac{c - a}{s - a}) = 2$
$(b - c) (c - a) = 2 {(s - a)}^{2}$ $\Rightarrow b^{2} + c^{2} = a^{2}$
The triangle is right angled