Determine which triangle is this

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In summary, the given equation and formulas can be used to determine the type of triangle in question, which has ex-radii denoted by ##r_1, r_2, r_3##. By rewriting the equation and using the formulas, it is possible to determine the type of triangle.
  • #1
utkarshakash
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Homework Statement


If in a ΔABC , [itex]\left( 1-\dfrac{r_1}{r_2} \right) \left( 1-\dfrac{r_1}{r_3} \right) = 2 [/itex]
then which triangle is it?

r1, r2, r3 denotes ex-radii.

The Attempt at a Solution



I get this condition

[itex]a^2+2b^2+2c^2+3bc=3ac+3ab [/itex]

Also

[itex] \sum tan \frac{A}{2} tan \frac{B}{2} = tan^2 \frac{A}{2} [/itex]

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.
 
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  • #2
utkarshakash said:

Homework Statement


If in a ΔABC , [itex]\left( 1-\dfrac{r_1}{r_2} \right) \left( 1-\dfrac{r_1}{r_3} \right) = 2 [/itex]
then which triangle is it?

The Attempt at a Solution



I get this condition

[itex]a^2+2b^2+2c^2+3bc=3ac+3ab [/itex]

Also

[itex] \sum tan \frac{A}{2} tan \frac{B}{2} = tan^2 \frac{A}{2} [/itex]

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?
 
  • #3
Ray Vickson said:
What do ##r_1, r_2, r_3## mean? Are they angles, sides, or what?

They are the ex-radii.
 
  • #5
  • #6
Pranav-Arora said:
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.
 

FAQ: Determine which triangle is this

What are the three types of triangles?

The three types of triangles are equilateral, isosceles, and scalene.

How can I determine which type of triangle I have?

To determine which type of triangle you have, you need to measure the lengths of all three sides. If all three sides are equal, it is an equilateral triangle. If two sides are equal, it is an isosceles triangle. If all three sides are different lengths, it is a scalene triangle.

Can a triangle have more than one type?

No, a triangle can only have one type. It is either equilateral, isosceles, or scalene.

What is the difference between a right triangle and an acute triangle?

A right triangle has one angle that measures 90 degrees, while an acute triangle has all angles measuring less than 90 degrees.

How do I find the area of a triangle?

The formula for finding the area of a triangle is A = 1/2bh, where A is the area, b is the base, and h is the height. You can also use the Heron's formula, which is A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter and a, b, and c are the lengths of the sides.

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