# 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

The number of distinct right triangles circumscribed to a given circle is infinite.

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.

• If we go on extending point B on +ve x-axis then there will be some point when line CB will become parallel to x-axis. So there should be finite no of triangles with integer lengths. – quintin Oct 5 '16 at 8:22
• No, no such point exist. As far on thé right point B is taken, BC will have a (negative) slope, even very small. – Jean Marie Oct 5 '16 at 8:59

Hint: draw a right angle, and inscribe a circle of prescribed radius within it, and try to draw a few lines tangent to the circle.