# No of right triangles possible for a given in-radius R with integer sides?

We know that for a given in-radius there exist at least one right triangle.

Is there any way we can find the no of distinct right triangles with integer sides (distinct if any of the three sides differ) possible with side lengths for a given in-radius.

• I've changed the title and the actual statement of the question to reflect your first paragraph (so that they refer to right triangles). If that is not what you meant, you should make that clear. – tomasz Oct 4 '16 at 16:13
• Math musing question in math magazine? – N.S.JOHN Oct 5 '16 at 1:55

Proof: consider circle with center $(1,1)$ and radius one.
Fix $A=(0,0)$, and take any point $B(b,0)$ on $Ox$ axis with any $b>2$. Draw a tangent to the circle that intersects axis $Oy$ in $C(0,c)$. Triangle $ABC$ is a solution.