Positive integers a<b<c are lengths of sides of a right triangle whose inradius is equal to gcd(a+1,b)^{2}. Find a,b,c.

One inradius formula is Area=radius*semiperimeter.

Plugging in the sides of a right triangle and simplifying yields radius = a*b/(a+b+c).

Now lets use a parameterization of the right triangle: u^2-v^2, 2uv, u^2+v^2. Plug this into the radius formula to get radius = v*(u-v).

This value is to be a perfect square.

Trying small values I quickly find v=1 and u=10 to give a solution.

a=2*1*10=20, b=10^2-1^2=99, c=10^2+1^2=101, and radius=1*(10-1)=9.

gcd(20+1,99)^2=3^2=9. These match, so the solution is verified. **a,b,c are 20,99,101**.

*Edited on ***February 18, 2024, 10:53 am**