Question

In triangle $ABC$, If $(1-\frac{r_1}{r_2})(1-\frac{r_1}{r_3})=2$, then the triangle is:

• A. Isosceles
• B. Equilateral
• C. Right angled
• D. Obtuse angled

Answer 1

The correct option is C Right angled

Given: $(1-\frac{r_1}{r_2})(1-\frac{r_1}{r_3})=2$
$\Rightarrow (1-\frac{\frac{\Delta }{(s-a)}}{\frac{\Delta }{(s-b)}})(1-\frac{\frac{\Delta }{(s-a)}}{\frac{\Delta }{(s-c)}})=2$
$\Rightarrow (1-\frac{(s-b)}{(s-a)})(1-\frac{(s-c)}{(s-a)})=2$
$\Rightarrow \left(\frac{(s-a)-(s-b)}{(s-a)}\right)\left(\frac{(s-a)-(s-c)}{(s-a)}\right)=2$
$\Rightarrow (b-a)(c-a)=2(s-a{)}^2$
$\Rightarrow (b-a)(c-a)=\frac{1}{2}(b+c-a{)}^2$
$\Rightarrow 2(bc-ab-ac+a^2)=b^2+c^2+a^2+2bc-2ab-2ac$
On solving, we have: $b^2+c^2=a^2$
Hence the triangle is right angled.