Given: Triangle ABC, ex radius r1:r2:r3 = 1:2:3

To find: Ratio of sides a:b:c

Solution:

Step 1: Understanding Exradii

Exradii are the radii of circles drawn tangent to one side of a triangle and to the extensions of the other two sides. The exradii are denoted as r1, r2, and r3, corresponding to the vertices A, B, and C, respectively.

Step 2: Using Exradii ratio to find sides ratio

The ratio of exradii r1:r2:r3 = 1:2:3 means that r1 is one-third of the sum of the three exradii, r2 is two-thirds of the sum of the three exradii, and r3 is the sum of the three exradii.

Let the semiperimeter of the triangle be s, then we have:

r1 = (s-a)
r2 = (s-b)
r3 = (s-c)

Using the given ratio of exradii, we have:

r1:r2:r3 = 1:2:3
(s-a):(s-b):(s-c) = 1:2:3

Let's assume a = x, b = y, and c = z. Then,

s = (a+b+c)/2 = (x+y+z)/2

Substituting this value in the above equation, we get:

((x+y+z)/2 - x):((x+y+z)/2 - y):((x+y+z)/2 - z) = 1:2:3

Simplifying this equation, we get:

(y+z-2x):(x+z-2y):(x+y-2z) = 1:2:3

Therefore, the ratio of sides a:b:c is 1:2:3.

Step 3: Conclusion

Hence, the ratio of sides of triangle ABC is 1:2:3.