hiii

Now let me answer the first one ..

Let the radius of the circle be r .

 

Now let find the side of triangle ...

let a be the side of triangle .. now in the equilateral triagle the centroid and the incenter are same

 

so we have 1/3(length of median) = r

 

Now length of median =  a² - (a/2)²    = ( 3)/2.a

 

therefore a = 2. 3.r

 

Now in a square the radius is half the side .. so here a = 2r

 

Now in a regular hexagon .. each side can be joined with the centre to make six equilateral triangles ..

 

Also the altitude dropped from center to each side is r.

 

Now if length of hexagon's side is a ..

 

then, we have ,  3/2a = r ... or a = 2/ 3.r.

 

Therefore the sides of equilateral triangle , square and regular hexagon are

2. 3r , 2r, 2/ 3 r respectively which are having a common ratio of 1/ 3

 

Now about the second question ...

dear we want to keep the rules of the game same for alll .. plss .. let us keep it that way only ...

 

cheers