Question

Prove that the in radius of right angled triangle with integer sides is an integer.

Answer 1

let side of right angle triangle is a,b and c which are integer
due to right angle triangle
b^2+c^2=a^2. {let C is right angle

now ,
if circle encribed then radius of circle,

let r is the radius of circle
all three radius divide triangle in three part ,
now sum of each part area =area of triangle
1/2r x a x r +1/2 x r x b + 1/2 x r x c=1/2bc
1/2r(a+b+c)=1/2bc

r=bc/(a+b+c)

let first integer side 3,4, and ,5

put here we find r=3 x 4 /(3+4+5)=12/12=1
hence this is correct to say that if all side gain integer then inradius is also integer