Determine which triangle is this

[utkarshakash]

Homework Statement

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.

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.

 

[Ray Vickson]

Homework Statement

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?

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.

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

 

[utkarshakash]

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

They are the ex-radii.

 

[SteamKing]

A little bit of context for those who are unfamiliar with the terms:

http://mathworld.wolfram.com/Exradius.html

 

[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.

http://mathworld.wolfram.com/Exradius.html

 

[utkarshakash]

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.

http://mathworld.wolfram.com/Exradius.html

Thanks. The formulas given in Wolfram were of great help.