# Determine which triangle is this

1. Dec 31, 2013

### utkarshakash

1. The problem statement, all variables and given/known data
If in a ΔABC , $\left( 1-\dfrac{r_1}{r_2} \right) \left( 1-\dfrac{r_1}{r_3} \right) = 2$
then which triangle is it?

r1, r2, r3 denotes ex-radii.

3. The attempt at a solution

I get this condition

$a^2+2b^2+2c^2+3bc=3ac+3ab$

Also

$\sum tan \frac{A}{2} tan \frac{B}{2} = tan^2 \frac{A}{2}$

where sigma denotes the sum of products of tan of the half-angles taken pairwise.

But it is difficult to figure out the type of triangle by just looking at the above two eqns.

Last edited: Dec 31, 2013
2. Dec 31, 2013

### Ray Vickson

What do $r_1, r_2, r_3$ mean? Are they angles, sides, or what?

3. Dec 31, 2013

### utkarshakash

They are the ex-radii.

4. Dec 31, 2013

### SteamKing

Staff Emeritus
5. Dec 31, 2013

### Saitama

Rewrite the given equation as

$$r_1r_2+r_2r_3+r_1r_3=r_1^2$$

The formulas mentioned in the following link are helpful to proceed further.